Elementary Lessons in Logic

Lesson 1: Definition and Sphere of the Science

The shortest definition of logic is that it is the Science of Reasoning. More often it is called the Science of the Laws of Thought, and some logicians like to be more exact still and call it the Science of the Formal, or Necessary, Laws of Thought. Before definitions like these can do us any good, we need to be clear about what the words in them mean. Once we are, we will probably find that there is not much difference between them.

By a Law of Thought we mean a certain uniformity: a way in which everyone thinks and reasons alike, and must think alike, as long as they don't make what we call mistakes, or contradict themselves, or fall into fallacies. The laws of thought are natural laws. We have no power to interfere with them, and they must never be confused with the man-made laws of a country, which people invent and can change. Every science is busy finding and describing the natural laws that the things it studies obey without exception. Astronomy, for example, studies the uniform way in which the heavenly bodies, and in fact all material things, tend to fall toward one another, the way a stone falls toward the earth, or to move around one another under the pull of that tendency. The universal law of gravitation is the natural law, or uniformity, that physical astronomy deals with.

In chemistry, the law of equivalent proportions describes the well-established fact that each chemical substance combines with every other only in certain fixed proportions. Exactly eight parts by weight of oxygen combine with one part of hydrogen to make water, for instance, and sixteen parts of oxygen combine with six parts of carbon to make carbonic acid gas (carbon dioxide) whenever an ordinary flame or fire burns. Wherever we can find uniformities or similarities, we are to that extent creating science and arriving at natural laws. But there may be, and there are, many things so changeable, so complicated and so uncertain that we can never be sure we have found laws they will always obey, and in those cases no science, in the proper sense of the word, is possible. There is no real science of human character, for example, because the human mind is too variable and too complicated to study that way. No two people are so alike that you can be sure one will act in every situation exactly as the other would. So it is impossible to sort people into classes in such a way that everyone in the same class will act the same way in any given situation.

But there is a science of human reason, or thought, separate from all the other workings of the mind that make up a person's character. That is because there are ways in which everyone does think and reason alike, and must. For example: if two things are identical with a third thing, they are identical with each other. This is a law of thought, and a very simple and obvious one, and we can notice two things about it:

1. Everyone thinks in agreement with it, and agrees that they do as soon as they understand what it means.

2. They think in agreement with it no matter what subject they are thinking about.

Suppose the things we are considering are these:

	London,
	The capital,
	The most populous city in Great Britain.

Since "the capital is identical with London," and "London is identical with the most populous city in Great Britain," it follows necessarily, in every mind, that "the capital is identical with the most populous city in Great Britain."

Or compare these three things:

	Iron,
	The most useful metal,
	The cheapest metal.

If we grant that "the most useful metal is iron" and "iron is the cheapest metal," it follows necessarily, in every mind, that "the most useful metal is the cheapest." These are two examples of the general truth that things identical with the same thing are identical with each other. We can call this a general, or necessary, form of thought and reasoning.

Now compare these three:

	The earth,
	Planets,
	Bodies revolving in elliptical orbits.

This time we can't say, as before, that "the earth is identical with the planets." It is identical with only one of the planets, so we say that "it is a planet." In the same way we can say that "the planets are bodies revolving in elliptical orbits," but they are only some of all the bodies that revolve that way. Even so, it follows that if the earth is among the planets, and the planets are among the bodies revolving in elliptical orbits, then the earth is among those bodies too.

The most elementary chemistry lets us argue the same way about these:

	Iron,
	Metals,
	Elementary substances.

Iron is one of the metals, and the metals are elements, that is, simple substances that cannot be broken down any further. The metals are among the elements, part of them, not the whole of them. It follows necessarily that "iron is one of the elementary substances." So we now have two examples of a fixed and necessary form of thought, one that holds true whatever things we apply it to. This form of argument can be put in several ways, and we will look at it closely in the lessons on the syllogism. One way to put it is: "part of a part is part of the whole." Iron is part of the class of metals, which is part of the class of elements, so iron is part of the class of elements.

So if I now offer another definition of logic, and say it is "the science of the necessary forms of thought," I hope the reader will see clearly what "necessary forms of thought" means. A form is something that can stay the same while the matter poured into it changes. Medals struck from the same dies have exactly the same form, but they can be made of different matter: bronze, copper, gold or silver. A building of exactly the same form could be built of stone or of brick. Furniture of exactly the same shape can be made of oak, mahogany, walnut and so on. We recognize the difference between form and substance in everyday objects without thinking about it, and in logic we can see in just the same way that the form of an argument is one thing, and the subject or matter that the argument happens to be about is quite another. We can almost show the form of our last two arguments to the eye, like this:

If we put three names inside the three pairs of brackets, marked X, Y and Z, so that the name in X comes under the name in Y, and the name in Y comes under the name in Z, then it follows necessarily that the first (X) comes under the last (Z).

Logic, then, is the science that works out and describes all the general forms of thought we must use as long as we reason validly. These forms are very many, though the principles they are built on are few and simple. This makes logic the most general of all the sciences. We need its help more often than the help of any other science, because each particular science deals with only some of the things that exist, and the sciences grow into very different and often unconnected branches of knowledge. Logic, though, deals with the principles and forms of thought that must be used in every branch of knowledge. It deals with the very origin and foundations of knowledge itself. It is true that the logical method used in one science may differ a little from the method used in another. But whatever its particular form, the method must be logical, and it must obey the laws of thought. In short, there is something all the sciences must have in common, something they must all obey as long as they hold to what is true and consistent, and the work of logic is to explain this common basis of all science.

One name that has been given to logic, the Science of Sciences, describes very well how far the reach of logical principles extends. People who work in the particular branches of knowledge seem to have known perfectly well that they owe allegiance to the highest of the sciences, because the names they have usually chosen say so. The word "logic" itself turns up in almost all the names recently adopted for the sciences. These are often casually called the "ologies," but they are really the "logics," since the "o" is only a connecting vowel or the end of the word before it. So geology is logic applied to explaining how the earth's crust was formed; biology is logic applied to the facts of life; psychology is logic applied to the nature of the mind; and the same goes for physiology, entomology, zoology, teratology (the study of abnormal growths), morphology, anthropology, theology, ecclesiology (the study of churches), thalattology (the study of the sea) and the rest. Each of these sciences openly admits, in its own name, that it is a special kind of logic. The name "logic" comes from the common Greek word λόγος (logos), which usually means a word, the sign and outward expression of a thought inside. But the same word was also used for the inward thought or reasoning that words express, and that is probably how later Greek writers on reasoning came to call their science ἐπιστήμη λογική, "logical science," and also τέχνη λογική, "logical art." The adjective λογική, used on its own, soon became the name of the science, in the same way that mathematics, rhetoric and other names ending in "-ic" started out as adjectives and were turned into nouns.

Footnote: Except philology, which is formed differently and means the love or study of words. If its name had been formed on the same plan, the science would be called "logology."

A good deal of rather trivial argument has gone into the question of whether logic should be thought of as a science only, an art only, or both at once. Sir William Hamilton even took the trouble to sort nearly every writer on logic according to which opinion they held. But it seems correct, and enough, to say this: logic is a science insofar as it simply investigates the necessary principles and forms of thought, and so teaches us what correct thinking consists of. It becomes an art when it sets about making rules to help people catch false reasoning. A science teaches us to know, and an art teaches us to do, and all the more advanced sciences lead to useful arts that go with them. Astronomy is the foundation of the art of navigating the ocean, and of arranging the calendar and keeping track of dates. Physiology is the basis of the art of medicine, and chemistry is the basis of many useful arts. In the same way, logic has been thought of as the basis of an art of correct reasoning, or investigation, which ought to teach the true method to be followed in every science. The famous British logician Duns Scotus, who lived in the thirteenth century, called logic the Science of Sciences, and he also called it the Art of Arts, which says everything about its preeminence. Others have defined it like this: "Logic is the art of directing the reason aright in acquiring the knowledge of things, for the instruction both of ourselves and others." Dr. Isaac Watts took this view of logic and called his well-known book "the Art of Thinking."

Still, it is fair to say that logic is more a science than an art, and here is the reason. Everyone picks up the ability and the habit of reasoning long before they have even heard the word "logic." They pick it up by the natural use of their minds, or by constantly and unconsciously copying other people. So in all very simple cases they follow the principles of the science correctly, but without knowing it. Yet the contradictory opinions and absurd fallacies put forward by people with no training show that this unaided use of the mind can't be trusted once the subject under discussion gets at all difficult or complicated. So studying logic cannot be useless. It explains the principles on which everyone has often reasoned correctly already, and it also points out the dangers of arguing wrongly. The reasoner becomes a correct reasoner on purpose, and learns to avoid the traps of fallacy on purpose. To say that people can reason well without the science of logic is about as true as saying they can stay healthy without medicine. So they can, as long as they are healthy; and reasoners can do without the science of reasoning, as long as they do reason correctly. But how many people can? A man might as well claim his body is immortal as claim his mind is infallible.

And if it is worth saying a few words in defense of logic as an art, because of misunderstandings that have arisen in the history of the subject, does anything need to be said in praise of it as a science? Everything great in science, in art or in literature is the work of the intellect. In the form of our bodies we are related to the animals, and in the part of us that perishes we are nothing but matter. What raises human beings above everything else on earth is a conscious intellect, the power of reasoning with general ideas. Who can say, then, that the nature and workings of this intellect are not very nearly the highest and most interesting subject we could ever study? No one could seriously deny the truth of Sir William Hamilton's favorite saying:

	In the world there is nothing great but man.
	In man there is nothing great but mind.

Questions and Exercises

1. What do we mean by a Law of Nature, and by a Law of Thought?

2. Explain the difference between the Form of Thought and the Matter of Thought.

3. In what sense can logic be called the Science of Sciences?

4. Where does the name "logic" come from?

5. How does a science differ from an art, and why is logic more a science than an art?

6. Can we say that logic is a necessary aid to correct reasoning, when people who have never studied logic reason correctly?

Lesson 2: The Three Parts of Logical Doctrine

We saw in the last lesson that logic is the Science of Reasoning, or the Science of the Necessary Laws of Thought, the laws we must follow if we are to argue consistently with ourselves and avoid contradicting ourselves. So argument, or reasoning, is strictly speaking the proper subject in front of us. But the most convenient and usual way to study logic is to look first at the parts that any argument must be made of. An architect has to know the materials of a building, and a mechanic the materials of a machine, before either can claim to understand how the thing is built. In the same way, it makes sense to describe the materials and instruments we work with in reasoning before we go on to the actual forms of argument.

Take a simple argument like the one in the last lesson:

	Iron is a metal,
	Every metal is an element,
	Therefore Iron is an element,—

We see that it is made up of three statements, or assertions. Each of them contains, besides some minor words, two nouns, or names of things, and the verb "is." In short, two names, or terms, joined by a verb make up an assertion, or proposition; and three such propositions make up an argument, which in this case is called a syllogism. So it is natural and convenient to describe terms first, since they are the simplest parts. Next we go on to the nature and the different kinds of propositions that are built out of terms. Then we will be ready to deal with the syllogism as a whole. These, then, are the three parts of logical doctrine.

We can say that the three parts of logic deal with terms, propositions and syllogisms. But it is just as true, or truer, to say that our real subject is the acts of mind that these forms of language point to. Logicians' opinions on this point have varied, or perhaps it is only the way they express themselves that has varied. Archbishop Whately says plainly that logic is entirely about language. Sir William Hamilton, Mr. Mansel and most other logicians treat it as being about the acts or states of mind that the words point to. Mr. J. S. Mill goes back to the things themselves that we are arguing about. So is the subject of logic language, thought, or objects? The simplest and truest answer is that in a certain sense it deals with all three. No process of reasoning can be explained or passed on to another person without words, so in practice we are limited to reasoning that has been put into the form of language. That means we will always be dealing with words, but only insofar as they are the instruments for recording and referring to our thoughts. The grammarian deals with language too, but he treats it simply as language, and his science stops once it has described and explained the forms, kinds and relations of words. Logic also deals with language, but only as the necessary sign of what the mind is doing.

Again, as long as we think correctly, we must think of things as they are. The state of mind inside us must match the state of things outside us whenever we get a chance to compare them. It is impossible, and we cannot even conceive, that iron should turn out not to be an elementary substance, if it is a metal and every metal is an element. We cannot suppose, and there is no reason to suppose, that the way our minds are built forces us to think of things differently from the way they are. If, then, we can assume that things really agree or differ just as correct logical thought leads us to believe they will, the views of the logicians I named do not seem impossible to reconcile. We deal with things insofar as they are the objects of thought, and we deal with language insofar as it is the embodiment of thought. If the reader keeps this explanation in mind, he will be spared some confusion later, when he reads different books on logic and finds that they vary enormously in their way of treating the subject, or at least in their way of putting it.

If logic, once it is put into language, has three parts—terms, propositions and syllogisms—then there must be just as many different kinds of thought, or operations of the mind. They are usually called:

	1. Simple apprehension.
	2. Judgment.
	3. Reasoning or discourse.

The first of these, Simple Apprehension, is the act of mind by which we simply become aware of something, or have a notion, idea or impression of it brought into the mind. The adjective "simple" means apart from other things, and "apprehension" means the mind's taking hold of something. So the name, or term, Iron instantly makes the mind think of a strong and very useful metal, but it does not tell us anything about it or compare it with anything else. The words sun, Jupiter, Sirius and St. Paul's Cathedral are terms too. They call up in the mind certain well-known objects, which stay in our memory even when they are not in front of our senses. In fact, a term like these is used simply as a stand-in for showing the actual things it names.

Judgment is a different action of the mind. It consists in comparing two notions or ideas of objects, which we got by simple apprehension, to find out whether they agree or differ. Clearly, then, we cannot judge or compare unless we are aware of two things, or have the notions of two things in the mind, at the same time. Suppose I compare Jupiter and Sirius. First I simply apprehend each of them. But when I bring them into comparison, I notice that they agree in being small, bright, shining bodies that rise and set and move around the sky at what looks like the same speed. On close examination, though, I notice that Sirius gives a twinkling, flickering light, while Jupiter shines steadily. Watching for longer shows that Jupiter and Sirius do not really move at the same steady speed: Jupiter shifts its position in the sky from night to night in a rather complicated way. If I widen the comparison to other heavenly bodies that I can apprehend, or see, at the same time, I will find a multitude of stars that agree with Sirius in giving a twinkling light and in staying perfectly fixed in position relative to one another. Meanwhile two or three other bodies can be seen that resemble Jupiter in giving a steady light and also in changing their place from night to night among the fixed stars. Through the action of judgment I have now formed in my mind the general notion of fixed stars, by mentally grouping together a number of objects that agree; and out of several other objects I have formed the general notion of planets. When I compare these two general notions, I find that they do not have the same qualities or appearances, and I state this in the proposition, "Planets are not fixed stars."

I have used the expression "General Notion" as if the reader already knew it well. But although philosophers have been using the expressions general notion, idea, conception, concept and so on for more than two thousand years, they have never managed to agree exactly on what the terms mean. One group of philosophers, called Nominalists, say it is all a matter of names. When we put together Jupiter, Mars, Saturn, Venus and the rest and call them planets, the common name is what binds them together in our minds. Others, called Realists, have claimed that besides these particular planets there really is something that combines the properties they all share, without any of the differences of size, color or motion that set them apart. Today, however, everyone agrees that nothing can physically exist that corresponds to a general notion. Anything that exists must be here or there, of this size or that size, so it would be one particular planet and not any planet whatever. The Nominalists seem just as wrong, though, because language, if it is to be of any use, must stand for something, and must correspond, as we have seen, to acts of mind. So if proper names raise in our minds the images of particular things, like the sun or Jupiter, general names should raise general notions.

The true view seems to be that of the philosophers called Conceptualists. They say that the general notion is the mind's knowledge of the common properties, or resemblances, of the things the notion takes in. So the notion planet really means the awareness, in anyone's mind, that there are certain heavenly bodies which agree in giving a steady light and in moving around the sky differently from the fixed stars. I should add, however, that many people, including Sir William Hamilton, would be counted as Nominalists and yet hold that the general name carries with it an awareness of the resemblance between the things it denotes. It is not easy to draw a precise line between this form of the doctrine and conceptualism, and the subject is too much a matter of debate to follow any further in this book.

It will become clear over these lessons that the whole of logic, and the whole of any science, consists in arranging the individual things we meet into general notions, or classes, and giving them suitable general names, or terms, so that our knowledge of them can be made as simple and as general as possible. Every properly formed general notion allows us to state general laws, or truths, about it. Of the planets, for example, we can affirm that they move in elliptical orbits around the sun from west to east, that they shine with light reflected from the sun, and so on. Of the fixed stars we can affirm that they shine with their own light, that they are incomparably farther away than the planets, and so on. All reasoning, we will find, comes from this faculty of judgment, which lets us discover and affirm that a large number of objects have similar properties, so that whatever is known of some of them can be inferred and asserted of the others.

It is in applying knowledge like this that we use the third act of mind, called discourse or reasoning. By it, starting from certain judgments, we are able to form a new judgment without going back to the real objects at all. If we know that iron comes under the general notion of metal, and that this notion comes under the still wider notion of element, then without examining iron any further we know that it is a simple substance that cannot be broken down, the kind chemists call an element. Or suppose we learn from one source that Neptune is a planet, and from another that planets move in elliptical orbits. We can join these two pieces of knowledge together in the mind and draw out the truth that Neptune moves in an elliptical orbit.

Reasoning, or Discourse, can therefore be defined as the progress of the mind from one or more given propositions to a proposition different from those given. The propositions we argue from are called Premises, and the one we draw from them is called the Conclusion. The conclusion is said to follow from the premises, or to be concluded, inferred or collected from them. The premises get their name because they are put forward, or put at the beginning (Latin præ, before, and mitto, I send or put). The heart of the process is gathering up the truth that the premises contain when they are joined together, and carrying it with us into the conclusion, where it takes shape in a new proposition, or assertion. We extract from the premises all the information that is useful for the purpose in hand—and that is all that reasoning does.

I have now pointed out the three parts of logical doctrine—Terms, Propositions, and Reasoning or Syllogism—into which the subject is conveniently divided, and we will go on to look at each of them. It is worth mentioning, though, that a fourth part has often been added, called Method, which deals with the arrangement of the parts of any composition.

It is sometimes said that method is to the syllogism what the syllogism is to the proposition, and what the proposition is to the term, and that a fourth division is needed to complete the doctrine of logic. In any case, this fourth part is certainly much less important and less clearly marked off than the other three, and everything that will be said about it is in Lesson 24.

Questions and Exercises

1. Name the parts that a syllogism is made of.

2. How far is it correct to say that logic is concerned with language?

3. What are the three acts of mind considered in logic? Which of them is more especially the subject of the science?

4. Can you state exactly what is meant by a general notion, idea, or conception?

5. How do the Nominalists, Realists, and Conceptualists differ in their opinions about the nature of a general notion?

6. What is the supposed fourth part of logic?

Lesson 3: Terms, and Their Various Kinds

In the last lesson we saw that every assertion or statement expresses the agreement or difference of two things, or of two general notions. So when we put an assertion into words, we need two kinds of words. We need words that draw the mind's attention to the things being compared, and we need words that give the result of the comparison, that is, whether the things agree or differ. The words that point out the things, or classes of things, are called Terms. The words that express the comparison are said to form the Copula. A complete assertion or statement therefore consists of two terms and a copula, and when it is put this way it forms a Proposition. Take the proposition "Dictionaries are useful books." The two terms are dictionaries and useful books. The copula is the verb are, and it expresses a certain agreement between the class of dictionaries and the class of useful books: the class of dictionaries forms part of the class of useful books. Here each term is only one or two words long, but a term can need any number of words to describe the notions or classes being compared. In the proposition "the angles at the base of an isosceles triangle are equal to each other," the first term takes nine words, and the second term takes four (equal to each other). There is no limit to the number of words that can go into a term.

A term gets its name because it forms one end (Latin terminus, an end or boundary) of a proposition, and strictly speaking it is a term only as long as it stands in a proposition. But we commonly use "term" or "name" to mean any noun, substantive or adjective, or any group of words that denotes an object of thought. As we will soon see, that object may be a single thing, a group of things, a quality of things, or a group of qualities. No one has defined a name or term better than Hobbes: "A name is a word taken at pleasure to serve for a mark, which may raise in our mind a thought like to some thought which we had before, and which, being pronounced to others, may be to them a sign of what thought the speaker had before in his mind."

Every term or name is made of words, but not every word can make a name by itself. We can't properly say "Not is agreeable" or "Probably is not true." Nothing can be asserted about a preposition, an adverb or certain other parts of speech, except that they are prepositions, adverbs and so on. Only a noun substantive, or a group of words used as a noun substantive, can form the subject, or first term, of a proposition. Only a noun substantive, an adjective, something equivalent to an adjective, or a verb can form the second term, or predicate. It is even open to question whether an adjective can ever form a term on its own. In "Dictionaries are useful," it may be said that the substantive things or books is understood in the predicate, so that the complete sentence is "Dictionaries are useful books." But since this is a disputed point, we will assume that words divide into two kinds, as follows.

Words that stand, or seem to stand, alone as complete terms, namely the substantive, the adjective and certain parts of a verb, are called categorematic words, from the Greek word κατηγορέω (katēgoreō), to assert or predicate.

The other parts of speech, such as prepositions, adverbs and conjunctions, can only form parts of names or terms. These are called syncategorematic words, because they have to be used with other words to make up terms (Greek σύν, syn, with, and κατηγορέω). We need not pay any more attention to syncategorematic words, except where they form part of categorematic terms.

Next we need to look at the different kinds of terms and their peculiarities, so that we get a clear idea of what they mean. The first division of terms is into singular or individual terms, and general or common terms. It is a very obvious division, but an important one. A Singular term is one that can denote only a single object, at least as long as it is used in exactly the same meaning. The Emperor of the French, the Atlantic Ocean, St Paul's, William Shakespeare and the most precious of the metals are all singular terms. All proper names belong to this class. John Jones is the name of many men, it is true, but it is not used to mean any one of those men you like. It is used to mean some single man. In short, it has a different meaning in each case, just as London, the name of our capital, has no connection in meaning with London in Canada.

General terms, by contrast, apply in the same sense, and equally well, to any one of an indefinite number of objects that resemble each other in certain qualities. Metal is a general name, for example, because it can be applied without preference to gold, silver, copper, tin, aluminum, or any of about fifty known substances. It is no more the name of one of these than of any other. In fact it is applied to any substance that has a metallic luster, cannot be decomposed, and has certain other qualities that chemists easily recognize. Nor is the number of substances in the class fixed: as new kinds of metal are discovered from time to time, they are added to it. Again, Mars, Jupiter, Saturn and the rest are singular terms, since each can denote only one planet. But the term planet is a general one, because it applies to as many bodies as may be found to revolve around the sun as the earth does.

We must be careful not to confuse general terms with collective terms. A collective term is the name of a number of things all joined together as one whole, like the soldiers of a regiment, the men of a jury, or the crew of a ship. A collective term is the name of all of them, but not of each. A general term, on the other hand, is the name of a number of things, but of each of them separately, or, to use the technical expression, distributively. Soldier, juryman and sailor are general names that may belong to John Jones, Thomas Brown and so on. But we can't say that John Jones is a regiment, or Thomas Brown a jury. Pointed out like this, the distinction is extremely obvious. But it can turn up in less obvious forms, and then it is likely to lead to faulty reasoning, as we will see in Lesson 20. Notice that we cannot divide terms into those that are general and those that are collective, because the same term will often be both, depending on how we look at it. Library is collective with respect to the books in it, but general with respect to the great number of different libraries, private or public, that exist. Regiment is a collective term with respect to the soldiers who make it up, but general with respect to the hundred different regiments, the Coldstream Guards, the Highland regiment, the Welsh Fusiliers and the rest, that make up the British standing army. Army, in its turn, is a collective whole, because it is made of a number of regiments organized together. Year is collective with respect to the months, weeks or days it consists of, but general because it is the name of 1869, or of 1870, or of any period marked by one revolution of the earth around the sun.

English does not always give us a convenient way to tell the general use of a term from its collective use. Latin expressed the difference exactly: omnes meant all distributively, and cuncti meant all taken together (it is a shortened form of conjuncti, joined together). In English, all men may mean any man, or all men together. Even the more exact word every is sometimes misused, as in the old proverb, "Every little makes a mickle" (a mickle is a large amount). Plainly, no little portion can make much all by itself. It can only do so when joined to other little portions.

A second important distinction is between concrete terms and abstract terms. It can't be described better than in the words of Mr. Mill: a concrete name is the name of a thing, and an abstract name is the name of a quality, attribute or circumstance of a thing. So red house is the name of a thing that physically exists, and it is concrete. Redness is the name of one quality of the house, and it is abstract. The word abstract means drawn from (Latin abstractus, from abstrahere, to draw away from). It tells us that the quality redness is thought of in the mind apart from all the other qualities that belong to the red house, or to any other red object. But although we can think of a quality by itself, we cannot suppose that the quality can exist physically apart from the matter in which it shows itself to us. Redness means either a notion in the mind, or whatever it is in red objects that excites that notion.

The reader should notice carefully that adjectives are concrete, not abstract. If we say that a book is useful, it is the book we apply the adjective useful to, and usefulness is the abstract noun that denotes the quality. In the same way, the adjectives equal, grateful, reverent and rational are names of things, and the abstract nouns that go with them are equality, gratitude, reverence and rationality. This distinction will become clearer when you read Lesson 5.

It is good practice to try to find pairs of matching concrete and abstract names. Animal has animality; miser, miserliness; old, agedness or old age; substance, substantiality; soap, soapiness; shrub, shrubbiness; and so on. But it does not follow at all that there is an abstract word for every concrete one. Table hardly has an abstract tabularity, and although ink has inkiness, we would not find an abstract for pen. Whether we have an abstract name or not depends on the accidents of the history of language, and as time and science move on, people keep inventing new abstract words.

Unfortunately, concrete and abstract names are often confused, and it is far from easy to keep their meanings apart. Relation, properly speaking, is the abstract name for the position two people or things stand in toward each other, and the people themselves are properly called relatives (Latin relativus, one who is related). But nowadays we constantly speak of relations, meaning the people themselves, and when we want to point to the abstract relation between them we have to invent a new abstract name, relationship. Nation has long been a concrete term, though judging by its form it was probably abstract at first. Yet the misuse of language has gone so far, especially in newspaper writing, that we hear of a nationality meaning a nation. Of course, if nation is the concrete, then nationality ought to be the abstract, meaning the quality of being a nation. In the same way, action, intention, extension, conception and a great many other names that are properly abstract are used loosely for the matching concrete words, namely act, intent, extent, concept and so on. Production is properly the condition or state of someone who is producing, or drawing something forth. But it has become confused with the thing produced, so that we constantly talk about the productions of a country, meaning its products. The logical terms Proposition, Deduction, Induction and Syllogism are all properly abstract words, yet they are used concretely for a Proposition, a Deduction, an Induction, a Syllogism. And it has to be admitted that logicians are nearly as bad as everyone else at confusing abstract and concrete terms. This misuse does a lot of harm to the language.

Another very obvious division of terms is between positive terms and negative terms. The usual way of describing the difference is to say that positive terms signify that a quality exists or is possessed, as in grateful, metallic and organic, while the matching negatives signify that the same qualities are absent, as in ungrateful, non-metallic and inorganic. Negative terms may be adjectives, like these, or substantives, either concrete or abstract. Ingratitude, inequality and inconvenience are abstract negative terms; individuals, unequals and so on are concrete negatives. We usually treat as negative any term with a negative prefix such as not, non, un or in. But a great many terms work as negatives without carrying any mark of it. Darkness is the negative of light or lightness, since it means the absence of light. Compound is the negative of element, since we give the name compound to whatever can be decomposed, and an element is what cannot be decomposed. In theory, every term has a matching negative, but it does not follow at all that the language supplies that term ready-made. Table has the matching adjective tabular, but there is no negative untabular to go with it. One person may be called a bookworm, but there is no negative for people who are not bookworms, because no one has felt the need for the expression. New negative terms are being invented all the time, perhaps faster than is good for us. When an idea is not mentioned often, it is better to express it with a phrase than to lengthen the dictionary with a newly made word.

In many cases, it seems, a negative term implies that some definite quality or fact is present. Inconvenience no doubt implies the absence of convenience, but it also implies the presence of real trouble or pain caused by that absence. Unhappiness is a negative term, but exactly the same notion is expressed by the positive term misery. The negative of healthy is unhealthy, but the positive term sickly does just as well. So whether a positive or a negative term is used for a particular notion seems to be more a matter of accident than anything else. All we can really say is that every positive term necessarily implies the possibility of a matching negative term, which is the name of all the things the positive name cannot be applied to. Whether that negative term has actually been invented is an accident of language; in logic we may assume it exists.

The reader should be warned not to assume that every term that looks negative because it has a negative prefix really is negative. The participle unloosed certainly looks like the negative of loosed, but the two words mean exactly the same thing: the prefix un is not really a negative here. Invaluable, again, does not mean having no value. It means so valuable that the value cannot be measured. And a shameless action can just as well be called by the positive term, a shameful action. No doubt other examples could be found.

Take great care not to confuse terms that express the presence or absence of a quality with terms that describe its degree. Less is not the negative of greater, because there is a third possibility, equal. The true negative of greater is not-greater, and that amounts to either equal or less. In the same way, we may say that disagreeable is not the simple negative of agreeable, because some things may be neither one nor the other, but simply indifferent to us. It would not be easy to say offhand whether every action that is not honest is dishonest, or whether there may be actions of an in-between character. The rule is this. Wherever the question is one of degree or quantity, a middle ground is possible, and the subject belongs to the science of quantity rather than to simple logic. Where the question is whether a quality is present or absent, there cannot be more than two alternatives, according to one of the Primary Laws of Thought, which we will look at in Lesson 14. In the case of quantity, we may call the extreme terms opposites: less is the opposite of greater, and disagreeable of agreeable. In the case of simple negation, we may call the terms negatives or contradictories. From a logical point of view, it makes no difference which of a pair of contradictory terms we treat as the positive and which as the negative. Each is the negative of the other.

Logicians have set apart from simple negative terms a class of terms called privative, such as blind, dead and so on. Such terms express that a thing has been deprived of a quality that it had before, or could have had, or usually does have. A man may be born blind, so that he never did see; but he has the organs that would have let him see if not for some accident. A stone or a tree could not have had the faculty of sight under any circumstances. No mineral substance can properly be said to die or to be dead, because it was never capable of life. But it may be called uncrystallized, because it could have been in the form of a crystal. So we apply a privative term to anything that lacks a quality it was capable of having, and a negative term to anything that lacks a quality and could not have had it. It is doubtful, though, whether this distinction can be properly carried through, and it does not matter very much.

It is also usual to divide terms into relative and absolute, that is, non-relative. The adjective absolute means whatever is "loosed from connection with anything else" (Latin ab, from, and solutus, loosed). Relative means whatever is carried, in thought at least, into connection with something else. So a relative term denotes an object that cannot be thought of without reference to some other object, or as part of a larger whole. A father cannot be thought of except in relation to a child, a monarch except in relation to a subject, a shepherd except in relation to a flock. So father, monarch and shepherd are relative terms, while child, subject and flock are their correlatives (Latin con, with, and relativus): the objects that are necessarily joined in thought with the first ones. The very meaning of father, in fact, is that he has a child; of monarch, that he has subjects; and of shepherd, that he has a flock. As examples of terms with no obvious relation to anything else, I might mention water, gas and tree. Nothing seems to me so habitually connected with water that we must think of it as part of the same idea. Gas, tree and a great many other terms also denote objects that have no notable or permanent relations of the kind that would earn the terms the name of relatives. So they may be considered absolute, or non-relative, terms.

The truth, though, is that everything really must have relations to something else: water to the elements it is made of, gas to the coal it is manufactured from, the tree to the soil it is rooted in. And by the very laws of thought, we cannot think of any thing, or class of things, except by separating it from the other existing things it differs from. I cannot use the term mortal without immediately splitting all existing or conceivable things into two groups, mortal and immortal. Metal, element, organic substance and every other term one could mention would necessarily imply that a matching negative term exists: non-metallic, compound, inorganic substance. In this respect, then, every term is certainly relative. Logicians, however, have been content to treat as relative terms only those that imply some special and striking kind of relation, arising from position in time or space, from the connection of cause and effect, and so on. It is in this special sense that the student should use the distinction.

Now that the most important kinds of terms have been explained, the reader should get thoroughly familiar with them by working through the exercises at the end of this lesson. For each of the terms given there, the reader is to decide:

1. Whether it is a categorematic or syncategorematic term.

2. Whether it is a general or a singular term.

3. Whether it is collective or distributive.

4. Whether it is concrete or abstract.

5. Whether it is positive, or negative, or privative.

6. Whether it is relative or absolute.

The next lesson will show in full that most terms have more than one meaning. One meaning may be general and another singular, one concrete and another abstract, and so on. So it is absolutely necessary for the reader first of all to choose one exact meaning of the term being examined. And in answering the questions, it is a good idea to state which way you are taking it. Take the word sovereign. We may first pick the meaning in which it is equivalent to monarch. In this meaning it is a general term, insofar as it is the name of any one of many monarchs, living or dead. But it is singular with respect to the people of any one country. It is clearly categorematic, concrete and positive, and obviously relative to the monarch's subjects.

Read Mr. Mill's chapter on Names, System of Logic, Book I. chap. 2.

Questions and Exercises

1. Define a name or term.

2. What is a categorematic term?

3. Explain the distinction between a collective and a general term.

4. Distinguish the collective and the distributive use of the word all in the following:

(1) Non omnis moriar (that is, I shall not all die).

(2) "All men find their own in all men's good, And all men join in noble brotherhood." Tennyson.

(3) Non omnia possumus omnes (that is, we cannot all do all things).

5. Which of the following are abstract terms? Act, ingratitude, home, hourly, homeliness, introduction, individuality, truth, true, trueness, yellow, yellowness, childhood, book, blue, intention, reason, rationality, reasonableness.

6. Define a negative term, and give the mark by which you can recognize one.

7. Distinguish a privative term from a negative term, and find some examples of privative terms.

8. Describe the logical character of each of the following terms, taking the precautions given earlier in this lesson.

Lesson 4: Of the Ambiguity of Terms

No part of logic is more genuinely useful than the part that deals with the ambiguity of terms: the uncertainty, and the variety, of the meanings that words carry. Nothing matters more for good habits of thinking and reasoning than a thorough knowledge of how imperfect language is. Comparatively few terms have one clear meaning and one meaning only. And whenever two or more meanings get mixed up without our noticing, we are bound to commit a logical fallacy. Suppose someone argues that "punishment is an evil," and that by the principles of morality "no evil is to be allowed even with the purpose of doing good." At first we might not see how to escape the conclusion that "no punishments should be allowed," since they cause evil. A little thought shows that the word evil is being used here in two completely different senses. In the first statement it means physical evil, or pain. In the second it means moral evil, or wrongdoing. Moral evil must never be committed, but it does not follow that physical evils must never be inflicted, because they are often the very means of preventing moral evil.

Here is another very convincing fallacy, which has been put forward in many forms: "A thoroughly benevolent man cannot possibly refuse to relieve the poor; and since a person who cannot possibly act otherwise than he does can claim no merit for his actions, it follows that a thoroughly benevolent man can claim no merit for his actions." By this kind of argument, the more virtuous a person became, and so the harder he found it to act wrongly, the less credit he would deserve. Everyone feels sure that the conclusion is false. But the cause of the fallacy can only be found by noticing that the words cannot possibly have a double meaning. In the first case they refer to the pull of moral motives, or good character. In the second they refer to circumstances entirely beyond a person's control, such as being forced by the law, lack of money, or lack of personal freedom. The more closely you study the subtle shifts in the meaning of common words, the more convinced you will be that the tools you have to use in every conversation and argument are dangerous ones. So I must ask you to pay close attention to this lesson.

Terms are called univocal when they can bring only one single, definite meaning to mind. They are called equivocal, or ambiguous, when they have two or more different meanings. Notice, though, that a term is not equivocal just because it can be applied to many objects, as long as it is applied to all of them in the same sense. Cathedral, for instance, is the name of St Paul's, of York Minster, and of the principal churches of Salisbury, Wells, Lincoln and a number of other cities. But it is not ambiguous, because these are all just different instances of the same meaning: they are all objects of the same description, or kind. The word cathedral is probably univocal, with only one logical meaning. The word church, on the other hand, is equivocal. Sometimes it means the building where religious worship takes place. Sometimes it means the body of people who belong to one sect or denomination and meet in churches. And sometimes "the church" means the clergy, as distinct from the laity (the ordinary members). So there is a clear difference in the sense in which the word is used at different times.

Univocal terms are found mostly in technical and scientific language. Steam-engine, gasometer (a large gas-storage tank), railway train, permanent way (the finished track bed of a railway), and a great many other technical names for distinct, familiar objects are univocal enough. In everyday life the names penny, mantelpiece, teacup, and bread and butter have a definite and single enough meaning. In chemistry, too, oxygen, hydrogen, sulphate of copper, alumina, lithia and thousands of other terms are very precise. Often the words themselves were invented quite recently, and their meanings were fixed exactly and kept unchanged. Every science has, or ought to have, a set of terms just as precise and certain in meaning (see Lesson 33). The names of individual objects, buildings, events or people are also usually quite certain and clear: Julius Caesar, William the Conqueror, the first Napoleon, St Peter's, Westminster Abbey, the Great Exhibition of 1851, and so on.

But however many univocal terms we can list, equivocal terms are astonishingly common. They include most of the nouns and adjectives we use every day in ordinary life. They are called ambiguous, from the Latin verb ambigo, to wander, hesitate or be in doubt. They are also called homonymous, from the Greek ὁμός (homos), same, and ὄνομα (onoma), name. Whenever someone uses equivocal words in a way that mixes up their different meanings, and so falls into error, we can say he commits the fallacy of Equivocation, in the logical sense of that name (see Lesson 20). In everyday life, though, we only say someone equivocates when he uses words knowingly and dishonestly, in a way designed to confuse the true meaning with the apparent one.

I will now describe the different kinds and causes of ambiguity in words, following in part the interesting chapters on the subject in Dr. Watts's Logic. First, we can distinguish three classes of equivocal words, depending on whether they are:

	1. Equivocal in sound only.
	2. Equivocal in spelling only.
	3. Equivocal both in sound and spelling.

The first two classes matter comparatively little, and they don't often lead to serious error. They produce what we would call trivial mistakes. When spoken aloud, for example, we can confuse the words right, wright and rite (a ceremony); also rein, rain and reign, might and mite, and so on. Partly because of careless pronunciation, people have been known to confuse the four words air, hair, hare and heir.

Words equivocal in spelling but not in sound include tear (a drop from the eye) and tear, pronounced "tare," meaning a rip in cloth; or lead, the metal, and lead, as in following someone else's lead. Such resemblances can cause little more than a moment's misunderstanding, so we will move straight on to the class of words that are equivocal both in sound and in spelling. I will divide these into three groups, according to whether the equivocation comes:

1. From the accidental confusion of different words.

2. From the transfer of meaning by the association of ideas.

3. From the logical transfer of meaning to analogous objects.

I. In the first group we put a certain number of curious but not very important cases. Here the ambiguity comes from the confusion of entirely different words. They came from different languages, or from different roots in the same language, but over time they took on the same sound and spelling. Take the word mean. It can mean what is medium or mediocre, from the French moyen and the Latin medius, which are related to the Anglo-Saxon mid, or middle. Or it can mean what is small-minded and base, and then it comes from the Anglo-Saxon gemæne, meaning "belonging to the mæne, or many": in short, whatever is common or vulgar. The verb to mean can hardly be confused with the adjective mean, but it comes from a third, separate root, probably related to the Sanskrit verb for "to think."

Here are some other examples of this accidental kind of ambiguity. Rent can be a money payment, from the French rente (from rendre, to give back). Or it can be a tear, the result of the action of rending. This second word is of Anglo-Saxon origin, and belongs to the large group of words beginning with r or wr that imitate, more or less closely, the sound of the action they name. Pounds, from the Latin pondus, a weight, gets confused with pound in the sense of a village pinfold, the pen where stray cattle were shut up, which comes from the Saxon pyndan, to pen up. Fell, a mountain, is a completely different word from fell, a skin or hide (a sense already old-fashioned by the 1870s). And pulse, a throb or beat, and pulse, meaning peas, beans or a soup made from them, though both come from Greek or Latin, are probably quite unrelated words. Curiously, gin in the sense of a trap or machine is a shortened form of engine, while gin the alcoholic drink is a corruption of Geneva, the place where the spirit was first made.

Some important cases of confusion have been found in grammar. One example is the confusion between the number one, which comes from an Aryan root (what we now call Indo-European) through the Latin unus, and the indefinite pronoun one (as in "one ought to do one's duty"). The pronoun is really a corrupted form of the French word homme, meaning man. To this day the Germans use man in this sense, as in man sagt, that is, "one says."

2. By far the largest share of equivocal words became equivocal through a transfer of meaning. The word moved from the thing it first named to some other thing that was regularly connected with it, so that the two became closely linked in our thoughts. In the language of Parliament, for example, "the House" means either the chamber where the members meet, or the body of members who happen to be gathered in it at any time. In the same way, the word church originally meant the building (κυριακόν, kyriakon, "the Lord's House") where any religious worshippers gather, but from there it has picked up a variety of meanings. It can mean a particular group of worshippers who regularly meet in one place, which is the sense it has in Acts 14:23. It can mean any body of people who hold the same beliefs and are joined in one organization, as in the Anglican, Greek or Roman Catholic Church. It is sometimes used to include the laity as well as the clergy. But perhaps more often the clergy and religious authorities of a sect or country are so strongly associated with the act of worship that they are called the church in a special sense, as if they were the church above all others. It is also quite clear that the word means something entirely different depending on whether it is used by a member of the Anglican, Greek, Roman Catholic, Scottish Presbyterian, or any other existing church.

The word foot has gone through several curious but very clear transfers of meaning. It first meant the foot of a person or an animal, and it is probably distantly related to the Latin pes, pedis and the Greek πούς, ποδός (pous, podos). Because the length of a foot is a natural rough measure of length, the word came to be used for a fixed measure of length. And because the foot is at the bottom of the body, the name was extended by analogy to the foot of a mountain or the feet of a table. By a further extension, any position, plan, reason or argument that we stand on and rely on is called our foot, or footing. The same word also means soldiers who fight on their feet, that is, infantry, and the measured unit of a line of verse, which has a definite length. These very different meanings are naturally connected with the original one, and we can see this from the fact that the Latin and Greek words for foot go through exactly the same series of ambiguities.

It would be a long job to trace out all the various, and often contradictory, meanings of the word fellow. Originally a fellow was one who follows another, that is, a companion. From there it came to mean the other one of a pair, as one shoe is the fellow of the other, or simply an equal, as when we say that Shakespeare "hath not a fellow." From the plain meaning of companion, it also came to mean, vaguely, a person, as in the question "What fellow is that?" But there is a curious mixture of scorn and affection in the word. In the 1870s, to call a man a mere fellow, or simply a fellow in a certain tone of voice, was to show him severe contempt. Change the tone of voice, or the words around it, even slightly, and it becomes one of the warmest and most affectionate names there is, as when we speak of a dear fellow or a good fellow. We can add the technical meanings of the word too, as in a Fellow of a college, or of a learned society.

Another good example of many different meanings growing from a single root is the word post. Originally a post was something posited, that is, placed firmly in the ground, such as an upright piece of wood or stone. That meaning survives in a lamp-post, a gate-post, a signal-post, and so on. A post was often used to mark a fixed spot of ground, as a mile-post does, so the word came to mean the fixed or appointed place where the post stood, as in a military post, or the post of danger or of honor. In the days of the Roman Empire, the fixed stations where horses were kept ready, so that people could travel fast, were called posts in this way. From there the whole system for carrying people or news came to be called the posts. The word still has exactly this meaning in most of Europe, and we still use it in post-chaise (a hired fast carriage of the horse-drawn era), post-boy and post-horse, and in postillion (the rider on one of the horses). A postal system for letters has now been running for about two centuries in England and other countries, so this is perhaps the meaning most closely tied to the word post today, and a number of expressions have arisen from it, such as post-office, postage, postal-guide, postman, postmaster, postal-telegraph and so on. Curiously enough, we now have iron letter-posts (the pillar boxes where letters are mailed), and in them the word post is brought back to exactly its original meaning.

I chose the words above for their curious variety of meanings, but I do not hesitate to claim that most common nouns have several meanings, some more and some fewer. Dr. Watts, in his Logic, suggests that the words book, bible, fish, house and elephant are univocal terms, but the reader will easily find ambiguities in each of them. Fish, for instance, means something very different in natural history from what it means to people who are not scientists. They include under it not only true fishes but shellfish, or mollusks, and the cetacea, such as whales and seals; in short, every swimming animal, whether it has the character of a true fish or not. Elephant, in a stationer's or bookseller's shop, meant a large size of paper in the 1870s rather than a large animal. Bible sometimes means a particular copy of the Bible, and sometimes the collection of works that makes up the Holy Scriptures. The word man is remarkably ambiguous. Sometimes it means man as distinguished from woman. At other times it is certainly used to include both sexes. In certain recent election cases, the lawyers could not decide whether the word man, as used in the Reform Act of 1867, ought or ought not to be read as including women. On other occasions man means a grown male as distinguished from a boy, and it often means someone who is emphatically a man, with a manly character. Now and then it is used the way groom is, to mean a servant, as in the proverb, "Like master, like man." At other times it means a husband in particular.

3. Third, among ambiguous words we must pick out those that get their different meanings in a somewhat different way, namely by analogy, or real resemblance. When we speak of a sweet taste, a sweet flower, a sweet tune, a sweet landscape, a sweet face, a sweet poem, we are plainly applying one and the same word to very different things. Something as concrete as a lump of sugar can hardly be compared directly with something as intellectual as Tennyson's poem "The May Queen." Even so, if the word sweet is to be called ambiguous, it is ambiguous in a different way from the words we looked at before. All these things are called sweet because of a particular kind of pleasure they give, and that pleasure cannot be described except by comparing it with sugar. In the same way we call a pain sharp, a disappointment bitter, a person's temper sour, the future bright or gloomy, an achievement brilliant. All these adjectives imply a comparison with the simplest kinds of bodily sensation. The adjective brilliant comes from the French briller, to glitter or sparkle, and it keeps this meaning fully when we speak of a brilliant diamond, a brilliant star and so on. But by what a subtle analogy we speak of a brilliant position, a brilliant achievement, brilliant talents, a brilliant style! We cannot speak of a clear explanation, indefatigable perseverance, a perspicuous style, or a sore calamity, without using in each of these phrases a double analogy to physical impressions, actions or events. (Perspicuous means clear, literally "see-through"; indefatigable means "that cannot be tired out.") We will see in Lesson 6 that we owe the creation of all the names for mental feelings and mental things to this process.

Read Watts's Logic, Chapter iv.

Locke's Essay on the Human Understanding, Book III, Chapters ix. and x.

Questions and Exercises

1. Define univocal terms, and suggest some terms that are perfectly univocal.

2. What other names are equivocal terms often called by?

3. Distinguish the three kinds of ambiguous terms, and find examples of each.

4. Distinguish the three causes that have produced the third and most important class of ambiguous terms.

5. Explain the ambiguity of any of the following terms. Assign each one to its proper cause, and trace, as far as you can, how each separate meaning grew out of the original meaning.

Lesson 5: Of the Twofold Meaning of Terms, in Extension and Intension

There is no part of logic I would more urgently ask the reader to pay attention to than the one I will try to explain clearly in this lesson. I mean the double meaning that most logical terms have: the meaning in extension, and the meaning in intension. I believe that a reader who once gets a thorough grasp of the difference between these two meanings, and learns to keep it always in mind, will have very little further trouble in the study of logic.

The meaning of a term in extension consists of the objects the term can be applied to. Its meaning in intension consists of the qualities that objects bearing that name must have. A simple example will make the distinction very clear. What is the meaning of the name "metal"? The first and most obvious answer is that metal means gold, or silver, or iron, or copper, or aluminum, or some other of the 48 substances that chemists know of and consider to be metallic in nature. These substances, then, make up the plain and ordinary meaning of the name, and this is the meaning in extension. But suppose someone asks why the name is given to all these substances and only to these. The answer has to be: because they have certain qualities that belong to the nature of a metal. So we cannot know which substances we may call metals, and which we may not, unless we know the qualities that are indispensable to the character of a metal. Chemists lay these down roughly as follows. (1) A metal must be an element, or simple substance, which cannot be decomposed or separated into simpler substances by any known means. (2) It must be a good conductor of heat and electricity. (3) It must have a great and peculiar power of reflecting light, known as metallic luster.

Footnote: It is doubtful whether all metals really do have metallic luster, and chemists would find it very hard to give any consistent account of how they use the name. But the statements in the text are true enough to serve as an example.

These properties are common to all metals, or nearly all, and they are what mark a metal out and distinguish it from other substances. So in a certain way they form the meaning of the name metal. This is called the meaning in intension, to distinguish it from the first kind of meaning.

Almost any other common name has a double meaning in the same way. "Steamship" denotes in extension the Great Eastern, the Persia, the Himalaya, or any one of the thousands of steamships that exist or have existed. In intension it means "a vessel propelled by steam-power." Monarch is the name of Queen Victoria, Victor Emmanuel (the king of Italy), Louis Napoleon (Napoleon III of France), or any one of a fair number of people who rule countries single-handed. The people themselves form the meaning in extension; the quality of ruling alone forms the intensive meaning of the name. Animal is, in extension, the name of any one of billions of creatures alive now, and of far greater numbers that have lived or will live. In intension it implies that all those creatures have a certain animal life and feeling, or at least the power of digesting food and exerting force, which are the marks of animal nature.

I should say here that logicians have described this distinction of extension and intension in several different ways of speaking. It is the special misfortune of logic to have too many names, or synonyms, for the same idea. So the intension of a term means the same as its comprehension, or connotation, or depth; and the extension means the same as its denotation, or breadth. This can be stated most clearly in the form of a scheme:

Of these words, denotation and connotation are used chiefly by Mr. J. S. Mill among modern writers on logic, and they suit the purpose very well. To denote is to mark down, and the name marks the things it can be applied or attached to; so metal denotes gold, silver, copper and so on. To connote is to mark along with (from the Latin con, together, and notare, to mark), so the connotation consists of the qualities described above, the qualities we imply a thing has when we use the name metal.

When we compare different but related terms, we can see that they differ in the quantity of their extension and of their intension. The term element, for example, has a greater extension of meaning than metal, because it takes in all the metals and other substances as well. But at the same time it has less intension of meaning, because the qualities of a metallic substance must include the qualities of an element, besides the other qualities peculiar to a metal. Or compare the terms metal and malleable metal. The second term plainly does not include the metals antimony, arsenic and bismuth, which are brittle substances. So malleable metal is a term of narrower meaning in extension than metal. But it is also deeper in intension, because it connotes, or implies, the quality of malleability on top of the general qualities of a metal. White malleable metal is narrower again in extension, because it does not include gold and copper. And I could go on narrowing the meaning with more qualifying adjectives until the term denoted only a single metal.

The reader will now see clearly that a general law of great importance connects the quantity of extension with the quantity of intension. It is this: as the intension of a term is increased, the extension is decreased. We must not suppose, though, that there is any exact proportion between how much one meaning grows and how much the other shrinks. If we join the adjective red to metal, for example, we narrow the meaning much more than if we join the adjective white, because there are at least twelve times as many white metals as red ones. Again, the term white man covers a considerable fraction of the extension of the term man, but the term blind man covers only a small fraction of it. So it is obvious that when we increase the intension of a term, we may decrease its extension by any amount.

In understanding this law we must carefully tell apart the cases where the intension of a term is only apparently increased from those where the increase is real. If I add the term elementary to metal, I shall not really change the extension of the meaning, because all the metals are elements, and the elementary metals are neither more nor less numerous than the metals. But then the intension of the term is really unchanged too, because the quality of being an element is already found among the qualities of a metal, and there is no point in stating it all over again. A quality that belongs without exception to the whole of a class of things is usually called a property of the class (see Lesson 12), and we cannot qualify or restrict a term by adding its own property.

This is a convenient place to note a distinction between terms that are connotative and terms that are non-connotative. Non-connotative terms are the ones that simply denote things, without implying any knowledge of their qualities. Mr. Mill thinks this distinction is very important, so it will be best to quote his own words:

Footnote: System of Logic, Vol. i. p. 31, 6th ed. Book I. Chap. ii. §5.

"A non-connotative term is one which signifies a subject only, or an attribute only. A connotative term is one which denotes a subject, and implies an attribute. By a subject is here meant anything which possesses attributes. Thus John, or London, or England, are names which signify a subject only. Whiteness, length, virtue, signify an attribute only. None of these names, therefore, are connotative. But white, long, virtuous, are connotative. The word white denotes all white things, as snow, paper, the foam of the sea, &c., and implies, or, as it was termed by the schoolmen, connotes the attribute whiteness. The word white is not predicated of the attribute, but of the subjects, snow, &c.; but when we predicate it of them, we imply, or connote, that the attribute whiteness belongs to them.

"All concrete general names are connotative. The word man, for example, denotes Peter, James, John, and an indefinite number of other individuals, of whom, taken as a class, it is the name. But it is applied to them, because they possess, and to signify that they possess, certain attributes.... What we call men, are the subjects, the individual Styles and Nokes; not the qualities by which their humanity is constituted. The name therefore is said to signify the subjects directly, the attributes indirectly; it denotes the subjects, and implies, or involves, or indicates, or, as we shall say henceforth, connotes, the attributes. It is a connotative name....

"Proper names are not connotative: they denote the individuals who are called by them; but they do not indicate or imply any attributes as belonging to those individuals. When we name a child by the name Paul, or a dog by the name Caesar, these names are simply marks used to enable those individuals to be made subjects of discourse. It may be said, indeed, that we must have had some reason for giving them those names rather than any others; and this is true; but the name, once given, is independent of the reason. A man may have been named John, because that was the name of his father; a town may have been named Dartmouth, because it is situated at the mouth of the Dart. But it is no part of the signification of the word John, that the father of the person so called bore the same name; nor even of the word Dartmouth to be situated at the mouth of the Dart. If sand should choke up the mouth of the river, or an earthquake change its course, and remove it to a distance from the town, the name of the town would not necessarily be changed."

I quote this in Mr. Mill's own words because, although it states very clearly the view held by Mr. Mill and many others, that view is nevertheless probably wrong. It confuses the connotation of a name with its etymological meaning, that is, with the circumstances that caused the name to be attached to a thing. Surely no one who uses the name England, and knows what it denotes, can be ignorant of the particular qualities and circumstances of the country, and these form the connotation of the term. To anyone who knows the town of Dartmouth, the name must imply that it has the circumstances that mark out that town at the present time. If the river Dart were destroyed or moved, the town would be changed to that extent, and the meaning of the name would change with it. The name would no longer denote a town on the Dart, but a town that used to be on the Dart, and it would be a mere accident of history that the form of the name no longer seemed to suit the town. In the same way, any proper name, such as John Smith, is almost meaningless until we know which John Smith is meant. It is true that the name alone connotes the fact that he is a Teuton (of Germanic stock) and a male. But as soon as we know exactly which individual it denotes, the name surely implies his particular features, appearance and character as well. In fact, the only way we can ever recognize a thing is by its particular qualities, features or circumstances. So no name could have any fixed meaning unless we attached to it, in our minds at least, enough of a definition of the kind of thing it denotes for us to know whether any given thing was denoted by it or not. If the name John Smith does not call up in my mind the qualities of John Smith, how am I to know him when I meet him? He certainly does not have his name written on his forehead.

Footnote: Further objections to Mr. Mill's views on this point will be found in Mr. Shedden's Elements of Logic. London, 1864. pp. 14, &c.

This, however, is a question that is far from settled. Since Mr. Mill is generally considered the best authority on the subject, the reader may do well to accept his opinion for the time being: that singular or proper names are non-connotative, and that all concrete general names are connotative. Abstract names, on the other hand, can hardly have any connotation at all. They already denote the attributes or qualities of something, so nothing is left over to form the connotation of the name. Mr. Mill does think that abstract names can often be considered connotative, as when the name fault connotes the attribute of hurtfulness as belonging to the fault. But if fault is a true abstract word at all, I would count hurtfulness as part of its denotation. I am inclined to think that faultiness is the abstract name, and that fault is usually used concretely, as the name of a particular action or thing that is faulty, or has faultiness. But the subject cannot be properly discussed here, and the reader should note Mr. Mill's opinion: abstract names are usually non-connotative, but may be connotative in some cases.

The subject of Extension and Intension can be followed up in Hamilton's Lectures on Logic, Lect. VIII., or in Thomson's Laws of Thought, Sections 48 to 52. It is also discussed at length in Spalding's Logic (Encyclopaedia Britannica, 8th ed.).

Questions and Exercises

1. Distinguish very carefully between the meanings in extension and in intension of these terms: quadruped, railway, human being, engine, mountain, Member of Parliament.

2. List the synonyms, or other names, that are used instead of extension and intension.

3. By what law is the quantity of extension connected with the quantity of intension? Show that the law holds true of the following series of terms:

(1) Iron, metal, element, matter, substance.

(2) Matter, organized matter, animal, man.

(3) Ship, steamship, screw-steamship, iron screw-steamship, British iron screw-steamship.

(4) Book, printed book, dictionary, Latin dictionary.

4. Distinguish between the connotation and the denotation of a term.

5. From the list of terms in question 8 of Lesson 3, pick out the terms that are non-connotative according to Mr. Mill's views.

6. Arrange the following terms in series as in question 3, placing each term of greater extension before a term of less extension. Point out which are the terms of greatest and of least intension in each series.

Lesson 6: The Growth of Language

We have seen that words become equivocal in at least three different ways. Different words can get confused by accident. A word can change its meaning because it is habitually associated with things other than the thing it first meant. And a word can be carried over, by analogy, to objects of a similar nature. We now need to look more closely at certain changes in language that come from that last cause, because they are going on all the time. In fact, we can almost watch language being created and extended, and for the logician the subject is a very instructive and important one. Two great processes, working in opposite directions, change language in this way:

1. Generalization, by which a name comes to be applied to a wider class of objects than before, so that the extension of its meaning is increased and its intension diminished.

2. Specialization, by which a name comes to be restricted to a narrower class, so that its extension is decreased and its intension increased.

The first change happens in the most obvious way. We notice a resemblance between some new object that has no name and some object we know well, and to express the resemblance we instinctively apply the old name to the new object. We are all well acquainted with glass, for example. If we come across some substance with the same glassy nature and look, we will be quick to call it a kind of glass, and if we meet this new kind of glass often enough, it will probably come to share the name equally with the old, original kind. The word coal has gone through a change like this. Originally it was the name of charked, or charred, wood, which was the main fuel in use five hundred years ago. When mineral coal came into use, it took its name from the older fuel, which it resembled more than it resembled anything else, but at first it was marked off as sea-coal or pit-coal. Now that it is by far the more common of the two, it has taken the plain name, and we mark off the charred wood as charcoal. Paper has gone through a similar change. It originally meant the papyrus used in the Roman Empire, and it was carried over to the new writing material made of cotton or linen rags, which came in at some quite uncertain date. The word character is interesting because of the way logic uses it. The Greek χαρακτήρ meant, strictly, a tool for engraving. By association it was carried over to the marks or letters engraved with that tool, and the word still keeps this meaning when we speak of Greek characters or Arabic characters, meaning figures or letters. But objects often have natural marks, signs or tokens that point them out just as well as artificial characters do, so the name was generalized. It now means any peculiar or distinctive mark or quality by which an object is easily recognized.

Changes of this kind are usually made by no one in particular and with no definite purpose. They come about through a sort of unconscious instinct among the many people who use the name. In the language of science, though, changes are often made on purpose, with a clear understanding of the generalization involved. In everyday life, for example, soap means only a compound of soda or potash with fat. Chemists, however, have deliberately widened the name to include any compound of a metallic salt with a fatty substance. So there are such things as lime-soap and lead-soap, and lead-soap is used in making ordinary diachylon plaster (a sticking plaster). Alcohol at first meant the product of ordinary fermentation, commonly called spirits of wine. Then chemists discovered that many other substances had a theoretical composition very like that of spirits of wine, and the name was adopted for the whole class. A long list of different kinds of alcohols can be found in Dr. Roscoe's lessons on chemistry, and the number of known alcohols can go on growing without limit as discoveries are made. Every one of the chemical terms acid, alkali, metal, alloy, earth, ether, oil, gas and salt can be shown to have gone through great generalizations.

The other sciences supply hardly fewer examples. A lens originally meant a lentil-shaped, or double convex, piece of glass, because that was the kind of glass opticians used most often. But as glasses of other shapes came into use alongside lenses, the name was extended to concave pieces of glass, and even to perfectly flat ones. The words lever, plane, cone, cylinder, arc, conic section, curve, prism, magnet, pendulum, ray, light and many others have been generalized in the same way.

In ordinary language we can see that even proper, or singular, names are often generalized. In Cicero's time, for example, a good actor was called a Roscius, after an actor of outstanding talent. The successor of Julius Caesar adopted the name Caesar as an official name of the Emperor, and it gradually came to mean the same thing as Emperor, so that in the present day the Kaisers of Austria and the Czars of Russia both take their title from Caesar. The abstract name Caesarism has even been formed, to express a kind of imperial system like the one Caesar set up. A king of Egypt built a famous tower on the island of Pharos, at the entrance to the harbor of Alexandria, and because of it lighthouses are called phares in French, and were once called pharos in English. And from the famous Roman general Quintus Fabius Maximus, anyone who avoids bringing a contest to a showdown is said to follow a Fabian policy.

In science, too, singular names are often extended, as when the fixed stars are called distant suns, or the companions of Jupiter are called his moons. There is even a theory, and a likely one, that all general names were created by this process of generalization in the early ages of human progress. Grasping general notions takes a higher intellect than grasping single, concrete things, so it seems natural that names should at first have stood for individual objects, and only afterward been extended to classes. We catch a glimpse of this process in the case of the Aboriginal Australians who had been used to calling a large dog Cadli. When horses were first brought into the country, they took up this name as the nearest description of a horse. Captain Cook tells a very similar story about the people of Otaheite (Tahiti). It can be objected, though, that some act of judgment must have been made before anyone could see that a name suited a particular thing. So it seems likely that specialization as well as generalization was at work at the very beginning of language, much as both are now.

Specialization is the exact opposite of generalization, and it is almost as important. It consists in narrowing the extension of a general name, so that it comes to be the name only of an individual, or of a small part of the original class. This is how we get the names we need for the many new tools, occupations and ideas we deal with as civilization advances. The name physician comes from the Greek φυσικός, natural, and φύσις, nature, so properly it means someone who has studied nature, especially the nature of the human body. It has become restricted, however, to people who use this knowledge for medical purposes, and those who investigate natural science have had to adopt the new name physicist. The name naturalist has been restricted in the same way, to people who study living things. The name surgeon originally meant a handicraftsman. It is a corrupted form of chirurgeon, which comes from the Greek χειρουργός, hand-worker. But it has long been specialized to the people who carry out the mechanical parts of the healing art.

Language is full of examples just as good. Minister originally meant a servant, someone who acted as the minor, or lesser, of another. Now it often means, specifically, the most important man in the kingdom. A chancellor was a clerk, or even a doorkeeper, who sat in a space closed off by bars, or cancelli, in the offices of the Roman Emperor's palace. Now it is always the name of a high official, or even the highest. A peer was an equal (Latin, Par), and we still talk about being tried by our peers; but now, through the strange accidents of language, it means the few who rank above the rest of the Queen's subjects. Deacon, Bishop, Clerk, Queen, Captain and General are all words that have been through a similar process of specialization. In words like telegraph, rail, signal and station, and many other words to do with new inventions, we can follow the change happening within a single lifetime.

One effect of specialization is that it very soon creates a difference between any two words that happen, for some reason, to be synonymous. Two or more words are called synonymous (from the Greek σύν, with, and ὄνομα, name) when they have the same meaning, as perhaps teacher and instructor do, or similarity and resemblance, beginning and commencement, sameness and identity, hypothesis and supposition, intension and comprehension. But the truth is that words commonly called synonymous are seldom perfectly so. There are almost always shades of difference in their meaning or use, which are explained in books like Crabb's English Synonyms. A process that Coleridge called desynonymization, and Herbert Spencer called differentiation, is always going on, and it tends to specialize one word of a synonymous pair to one meaning and the other to another. Wave and billow, for example, originally meant exactly the same physical effect. But poets have now taken over the word 'billow,' while wave is used mainly in practical and scientific matters. Undulation is a third synonym, and in time it will probably become the only scientific term for a wave. Cab was originally just a short form of cabriolet, and so meant much the same thing, but it is now specialized to mean almost nothing but a hackney cab, a cab for hire. In America, car is coming to be restricted to the meaning of a railway car.

We may note that it is a logical defect in a language to have a great many synonymous terms. We get into the habit of using them interchangeably, without being sure that they are free of ambiguities and hidden differences of meaning. English suffers especially from this inconvenience, because it has a complete series of words from Greek or Latin roots that are nearly synonymous with other words of Saxon or French origin. The same statement can, in fact, be put into either Saxon or classical English, and, as Whately has rightly remarked, we often seem to prove a statement merely by repeating it in different words. The wealth and variety of words may add to the rhetorical power of the language, but it also sets traps for all kinds of fallacies. (See Lessons 20 and 21.)

Besides generalization and specialization, a vast number of additions and changes come into language through the analogical, or metaphorical, extension of the meaning of words. No doubt this change could be said to be a kind of generalization, since there must always be some resemblance between the new and the old uses of the term. But the resemblance is often a very distant and obscure one, the kind we would call analogy rather than identity. Every word used metaphorically, or as a comparison, is a case of this process of extension. The name metaphor comes from the Greek words μετά, over, and φέρειν, to carry, and apparently expresses the carrying over of a word from its ordinary use to a special one. The old comparison of a ruler to the pilot of a ship, for example, has given rise to many metaphors, as when we say the Prime Minister is at the Helm of the State. In fact the word governor, with everything derived from it, is one result of this metaphor, since it is just a corrupted form of gubernator, steersman. The words compass, polestar, ensign, anchor and many others connected with navigation are constantly used metaphorically. Horses and hunting give us another set of metaphors: taking the reins of government, overturning the government, taking the bit between the teeth, the Government Whip, being heavily weighted, and so on. No doubt we could show that every other familiar occupation in life has supplied its own stock of metaphors.

It is easy to show, however, that this process does not only go on consciously today. It must have been at work throughout the history of language, and we owe to it almost all, and probably all, the words that express refined mental or spiritual ideas. The very word spirit, now the most refined and immaterial of ideas, is only the Latin spiritus, a gentle breeze or breathing. Inspiration, esprit (or wit) and many other words come from this metaphor. It is a truly curious fact that almost all the words in different languages for mind or soul imply the same comparison with breath. Soul is from the Gothic root meaning a strong wind or storm. The Latin words animus and anima are thought to be connected with the Greek ἄνεμος, wind. ψυχή certainly comes from ψύχω, to blow. πνεῦμα, air or breath, is used in the New Testament for Spiritual Being. And our word ghost has been said to have a similar origin.

Almost all the terms used in mental philosophy, or metaphysics, for the actions and phenomena of the mind come in the end from metaphors. Apprehension is putting out the hand to take hold of something; comprehension is taking things together in a handful; extension is spreading out; intention, bending toward; explication, unfolding; application, folding to; conception, taking up together; relation, carrying back; experience is going thoroughly through a thing; difference is carrying apart; deliberation, weighing out; interruption, breaking between; proposition, placing before; intuition, seeing into. The list could be extended almost without end. Our English name for reason, the understanding, obviously contains some physical metaphor that has never been fully explained, and the Latin intellect contains a metaphor too.

Every sense gives rise to words with refined meanings. Sapience, taste, insipidity and gout (in the sense of taste or relish) come from the sense of taste, and sagacity from the dog's extraordinary power of smell. But the sense of sight is by far the sharpest and most intellectual, so it gives rise to the larger part of language. Clearness, lucidity, obscurity, haziness, perspicuity and countless other expressions come from this sense.

It is truly astonishing how much power language has, through generalization, specialization and metaphor, to create many words from one single root. Professor Max Müller has given a remarkable example in the root spec, which means sight. It appears in the Aryan (Indo-European) languages: in the Sanskrit spas, in the Greek σκέπτομαι (with the consonants swapped round), in the Latin specio, and even in the English spy. Here is an incomplete list of the words developed from this one root: species, special, especial, specimen, spice, spicy, specious, speciality, specific, specialization, specie (gold or silver coin), spectre, specification, spectacle, spectator, spectral, spectrum, speculum, specular, speculation. The same root also combines with various prefixes, and so we get a series of words: suspect, aspect, circumspect, expect, inspect, prospect, respect, retrospect, introspection, conspicuous, perspicuity, perspective. Each of these, in turn, has a number of derivatives attached to it. From suspect, for example, we get suspicion, suspicable, suspicious, suspiciously, suspiciousness. By my own count there are at least 246 words in all, used at one time or another in the English language, that undoubtedly come from the one root spec.

J. S. Mill's Logic, Book IV. Chap. v. On the Natural History of the Variations in the Meanings of Terms.

Archbishop Trench, On the Study of Words.

Max Müller, Lectures on the Science of Language.

Questions and Exercises

1. Trace the generalization or specialization that has taken place in any of the following words: kind, genus, class, species, order, rank, Augustus, president, speaker, Utopia, rock, Commons, doctor.

2. Point out metaphors that come from the notions of weight, straightness, rock and wind.

3. Distinguish as precisely as you can the meanings of the following synonyms: sickness, malady; mud, mire; confutation, refutation; boundary, limit; mind, intellect; recollection, reminiscence; procrastination, dilatoriness; converse, reverse, obverse, inverse.

4. Make lists of all the words derived from any of the following roots:

(1) Tendere, to stretch, as in intention, attention.

(2) Ponere, to place, as in position, supposition.

(3) Genus, tribe or kind, as in genus, generation.

(4) Munus, gift, as in remuneration, common (Latin, Communis).

(5) Modus, shape or fashion, as in mood, moderate.

(6) Scribere, to write, as in scribe, inscription, describe.

(7) Capere, to take, as in deception, incipient.

Lesson 7: Leibnitz on Knowledge

When we study terms, we need to understand clearly what it takes to have a perfect notion of what a term means. When someone uses a name like monarch, or civilization, or autonomy, it points the mind toward some thing or some idea, and if possible we ought to have perfect knowledge of that thing or idea before we use the word. So what does this perfect knowledge consist of? What are its necessary characters? The famous mathematician and philosopher Leibniz tried to answer this question in a short treatise, or tract, first published in 1684. Several recent books on logic base what they say on the subject on this tract, and Mr. Baynes has added a complete translation of it at the end of his translation of the Port Royal Logic. Leibniz's own remarks are not always easy to understand, so I will not stick to his exact words. Instead I will try to give the simplest possible statement of his views, as Dr. Thomson and Sir William Hamilton have interpreted them.

Knowledge is either obscure or clear; either confused or distinct; either adequate or inadequate; and lastly, either symbolical or intuitive. Perfect knowledge must be clear, distinct, adequate and intuitive. If it falls short in any one of these respects, it is more or less imperfect. So we can classify knowledge as in the following scheme:

A notion, which simply means our knowledge of a thing, is obscure when it does not let us recognize the thing again and tell it apart from everything else. We have a clear notion of a rose, and of most common flowers, because we can recognize them with certainty and don't mix them up with each other. We also have a clear notion of our close friends, and of people we meet regularly, because we recognize them whenever we see them, with complete certainty and without hesitating. It is said that a shepherd, with practice, gets a clear notion of each sheep in his flock, so that he can pick out any one of them on its own. In the same way, a keeper of hounds learns the name and character of each hound, while other people have only an obscure idea of the hounds in general and could not tell one from another. But the geologist cannot give a clear idea of what sandstone, or conglomerate, or schist, or slate, or trap rock (a dark volcanic rock) consists of. Rocks vary endlessly in degree and character, and it is often barely possible to say whether a rock is sandstone or conglomerate, schist or slate, and so on. Among the lowest forms of life, the naturalist hardly has a clear notion of animal life as distinct from plant life: it is often hard to decide whether a protophyte (a simple one-celled organism) should be classed with the animals or with the plants.

Clear knowledge, in turn, is confused when we cannot pick out the parts and qualities of the thing we know, and can only recognize it as a whole. Anyone knows a friend instantly and could tell him apart from everyone else, yet would usually find it impossible to say how he knows him, or by what marks. He could not describe his friend's figure or features except in the very roughest way. Someone with no practice in drawing who tries to sketch even a familiar animal like a horse or a cow soon finds that he has only a confused notion of its shape, while an artist has a distinct idea of the shape of every limb. A chemist has a distinct as well as a clear notion of gold and silver. He can not only tell with certainty whether a piece of metal really is gold or silver, but can also name and describe exactly the qualities by which he knows it, and could, if necessary, list a great many other qualities as well. We have a very distinct notion of a chessboard, because we know it consists of 64 squares. All our ideas of geometrical figures, such as triangles, circles, parallelograms, squares, pentagons, hexagons and so on, are, or ought to be, perfectly distinct. But when we talk about a constitutional government, or a civilized nation, we have only the vaguest idea of what we mean. We cannot say exactly what it takes to make a government constitutional without also taking in governments we don't mean to include; and the same goes for civilized nations. These terms have meanings that are neither distinct nor clear.

Notice that no simple idea, such as the idea of the color red, can be distinct in the sense meant here, because nobody can analyze the color red or describe to someone else what it is. A person who has been blind from birth cannot be brought to imagine it, and the only way we can define what it is like is to put an actual red object in front of someone's eyes. The same is generally true of all simple sensations, whether tastes, smells, colors or sounds. So these may be known clearly, but not distinctly, in the sense Leibniz gives that word.

It is not easy to explain the difference Leibniz meant by the names adequate and inadequate. He says, "When everything which enters into a distinct notion is distinctly known, or when the last analysis is reached, the knowledge is adequate, of which I scarcely know whether a perfect example can be offered —the knowledge of numbers, however, approaches near to it."

So to have adequate knowledge of things, we must be able to pick out not only the parts that make up our notion of a thing, but also the parts that make up those parts. For instance, we might be said to have an adequate notion of a chessboard. We know it is made up of 64 squares, and we know each of those squares distinctly, because each one is made up of 4 equal straight lines joined at right angles. Even so, we cannot be said to have a distinct notion of a straight line, because we cannot really define it, or break it down into anything simpler. To be completely adequate, our knowledge would have to allow analysis after analysis, forever, and that would make adequate knowledge impossible. But, as Dr. Thomson remarks, we can count any knowledge as adequate if it carries the analysis far enough for the purpose we have in mind. An engineer, for instance, has adequate knowledge of a machine if he knows not only its various wheels and parts, but also what those parts are for, what they are made of, what shapes they have and how they act, provided that he also knows all the mechanical properties of the materials and all the geometrical properties of the shapes that could affect how the machine works. But nobody expects him to go further still and explain why iron or wood of a particular quality is strong or brittle, why oil works as a lubricant, or what axioms the principles of mechanical forces rest on.

Last of all, we must notice the very important distinction between symbolical and intuitive knowledge. Going by the original meaning of the word, intuitive would mean what we gain by seeing (from the Latin intueor, "to look at"). Any knowledge we have directly through the senses, or that comes straight into the mind, is called intuitive. So we may learn intuitively what a square or a hexagon is, but hardly what a chiliagon, a figure with 1000 sides, is.

We could not tell the difference by sight between a figure with 1000 sides and a figure with 1001 sides. Nor can we picture any such figure completely in the mind. We know it only by name, or symbolically. All large numbers, like the ones that give the speed of light (186,000 miles per second) or the distance of the sun (91,000,000 miles), are known to us only by symbols, and they are beyond the power of our imagination.

We know infinity in a similar way. So we can become acquainted, with the intellect, with something our senses could never tell us about. We also speak of nothing, of zero, of what contradicts itself, of the non-existent, and even of the unthinkable or inconceivable. These words stand for things that can never be pictured in the mind, much less perceived intuitively through the senses; they can only be handled in a purely symbolical way.

In arithmetic and algebra we deal mostly with symbolical knowledge alone. When we work through a long arithmetic question or an algebra problem, we don't need to stop at each step and picture to ourselves what the numbers and symbols mean. Algebra teaches us that if we multiply the sum of two quantities by their difference, we get the difference of their squares; in symbols, $(a+b)(a-b)=a^2-b^2$. It is easy to see that this is true, like this:

$$\begin{array}{c}\hfill a+b\\ \hfill a-b\\ \hfill a^2+ab\\ \hfill -ab-b^2\\ \hfill a^2+0-b^2.\end{array}$$

Here we are working in the dark, symbolically. We use the letters a and b according to certain fixed rules, without knowing or caring what they mean. And whatever meaning we later give to a and b, we can be sure the process still holds and the conclusion is true, without going over the steps again.

In geometry, though, we argue by seeing intuitively that each step is true. We actually hold a picture of the figures in our mind, and satisfy ourselves that the figures really do have the properties required. So the algebraic truth shown above in symbols can easily be proved true of lines, and of the rectangles formed by those lines, as a corollary of the 5th Proposition of Euclid's Second Book.

There is a lot to be said about the advantages of the intuitive and the symbolical methods compared with each other. The symbolical method is usually far less laborious, and gives the most widely applicable answers. But it seldom or never gives the same command and understanding of a subject as the intuitive method does. That is why the study of geometry is always indispensable in education, even though the same truths can often be proved more easily by algebra. It is Newton's special glory that he managed to explain the motions of the heavenly bodies by the geometrical, or intuitive, method, while the greatest of his successors, such as Lagrange and Laplace, have dealt with these motions with the help of symbols.

What is true of mathematics applies to every kind of reasoning. Words are symbols just as much as A, B, C or x, y, z are, and it is possible to argue with words without being aware of what they mean at all. Suppose I say that "selenium is a dyad element, and a dyad element is one capable of replacing two equivalents of hydrogen." Nobody who knows no chemistry will be able to attach any meaning to these terms, and yet anybody will be able to conclude that "selenium is capable of replacing two equivalents of hydrogen." Such a person argues in a purely symbolical way. In the same way, whenever we use words in everyday life without having their full and exact meaning in mind at that moment, we have symbolical knowledge only.

There is no worse habit for a student or a reader to fall into than accepting words in place of a knowledge of things. Reading a book of natural history about Infusoria, Foraminifera, Rotifera and the like is perhaps worse than useless if those names do not bring clear images to mind. And a student who has not watched experiments, and examined the substances with his own eyes, will not get much out of books on chemistry and physics (natural philosophy). He will meet hundreds of new terms in them, and to him they will be nothing but empty and confusing signs. For this reason we should never miss a chance to get to know, through our own senses, the forms, properties and changes of things. That way, the language we use can be used intuitively as far as possible, and we can be saved from the absurdities and fallacies we might otherwise fall into. In short, we should follow Bacon's advice, ipsis consuescere rebus: to accustom ourselves to things themselves.

Hamilton's Lectures on Logic. Lect. IX.

Baynes' Port Royal Logic. Part I. Chap. 9, and Appendix.

Questions and Exercises

1. What are the characters of perfect knowledge?

2. Describe the kind of knowledge we have of each of the following notions or objects: A syllogism. Electricity. Motion. A triangle. Eternity. The weight of the earth (5852 trillions of tons, counting a trillion, as the British did then, as a million million million). The color of the sky.

3. Explain exactly what you mean by intuitive knowledge.

Lesson 8: Kinds of Propositions

A term standing alone cannot express a truth. All it does is direct the mind to some object, or class of objects, about which something may be affirmed or denied; the term itself affirms or denies nothing. "Sun," "air," "table" each call up an object of thought in every mind, but we cannot say that "sun is true," or "air is mistaken," or "table is false." Before words or terms can express the reasoning acts of the mind that we call true or false, we have to join them into sentences, or propositions. "The sun is bright," "the air is fresh," "the table is unsteady" are statements that may be true or may be false, and in any circumstances we can at least ask whether they are true. Now, just as the logical term was defined as any combination of words expressing an act of simple apprehension, a logical proposition is any combination of words expressing an act of judgment. In short, the proposition is the result of an act of judgment put into the form of language.

What the logician calls a proposition, the grammarian calls a sentence. But although every proposition is a sentence, we must not suppose that every sentence is a proposition. There are in fact several kinds of sentence that are more or less distinct from a proposition: the Sentence Interrogative, or Question; the Sentence Imperative, or Command; the Sentence Optative, which expresses a wish; and the Exclamatory Sentence, which expresses an emotion of wonder or surprise. These kinds of sentence can perhaps be turned, by a more or less roundabout way of speaking, into the form of a Sentence Indicative, which is the grammatical name for a proposition. But until that has been done they have no proper place in logic, or at least no place that logicians have so far explained well enough.

The name proposition comes from the Latin words pro, before, and pono, I place. It means laying or placing before someone the result of an act of judgment. Every act of judgment, or comparison, must involve the two things being compared. So every proposition will naturally have three parts: the two terms, or names, standing for the things compared, and the copula, or verb, showing the connection between them as it was found in the act of judgment. Take the proposition "Gold is a yellow substance." It expresses an agreement in color between gold and certain other substances already called yellow. Gold and yellow substance are clearly the two terms, and is is the copula.

The first term of a proposition is always called the subject, because it stands for the underlying matter, so to speak, about which something is asserted (from the Latin sub, under, and jactum, laid). The second term is called the predicate, which simply means what is affirmed or asserted. This name comes from the Latin prædicare, to assert, which also gave French its word prédicateur, worn down in English into our "preacher." Don't confuse this Latin verb with the rather similar prædicere, which has an entirely different meaning: to predict, or foretell. I strongly suspect that newspaper writers and others who pedantically use the verb "to predicate" sometimes fall into this confusion and really mean to predict. In any case, a purely technical term like predicate should not be dragged into ordinary language without need, when there are so many other good words that could be used instead. This and every other technical term of science should be kept to its proper scientific use. Neglecting that rule damages the language of everyday life and the language of science at the same time.

Propositions fall into two kinds, depending on whether they make their statement conditionally or unconditionally. The proposition "If metals are heated they are softened" is conditional. It does not make an assertion about metals in general, only about metals in the circumstance of being heated. Any circumstance that has to be granted or supposed before the assertion applies is a condition. Conditional propositions are of two kinds, Hypothetical and Disjunctive, but it is best to leave them for a later lesson (Lesson 19). For some time to come we shall be concerned only with unconditional propositions. These are usually called Categorical Propositions, from the Greek verb κατηγορέω (kategoreo, to assert or affirm).

The following diagram gives a convenient picture of the classification of sentences and propositions as far as we have gone:

Now we need to look carefully at the different kinds of categorical propositions. They are classified according to quality and according to quantity. As regards quality they are either affirmative or negative; as regards quantity they are either universal or particular.

An affirmative proposition is one that asserts a certain agreement between the subject and the predicate, so that the qualities or attributes of the predicate belong to the subject. The proposition "gold is a yellow substance" states that gold agrees with other yellow substances in such a way that we know it has the color yellow, along with whatever qualities are implied in the name substance. A negative proposition, on the other hand, asserts a difference or disagreement, so that some at least of the qualities of the predicate do not belong to the subject. "Gold is not easily fusible" denies that the quality of being easily melted belongs to gold.

Propositions are also divided according to quantity into universal and particular propositions. If the proposition affirms that the predicate belongs to the whole of the subject, it is a universal proposition. An example is "all metals are elements," which affirms that the quality of being undecomposable, or simple in nature, is true of all metals. But if we say "some metals are brittle," the quality of brittleness is affirmed only of some indefinite portion of the metals, and nothing in the proposition lets us be sure that any particular metal you might name is brittle. The name particular comes from the diminutive of the Latin pars, a part, so you would expect it to mean a small part. In logic, though, it must be carefully understood to mean any part at all, from the smallest fraction up to nearly the whole. Particular propositions do not include cases where a predicate is affirmed of the whole of the subject, or of none of it, but they include every case between those limits. So we can count all propositions like the following as particular:

A very few metals are less dense than water.

Most elements are metals.

Many of the planets are comparatively small bodies.

Not a few distinguished men have had distinguished sons.

The reader must pay careful attention to a rather subtle point, explained further on: a particular proposition, although it asserts the predicate only of a part of the subject, does not deny that it is true of the whole.

Aristotle, in fact, held that there were four kinds of proposition altogether as regards quantity, namely:

	Proposition:
	    Universal.
	    Particular.
	    Singular.
	    Indefinite.

A singular proposition is one that has a singular term for its subject, as in:

	Socrates was very wise.
	London is a vast city.

But we can fairly treat a singular proposition as a universal one, because it clearly refers to the whole of its subject, which in this case is one single individual thing.

Indefinite, or indesignate, propositions are those that carry no mark of quantity at all, so that the wording gives us no way of judging whether the predicate applies to the whole of the subject or only to part of it. "Metals are useful" and "comets are subject to the law of gravitation" are indefinite propositions. In reality, however, such propositions have no distinct place in logic at all, and the logician cannot properly deal with them until their true and precise meaning has been made clear. The predicate must be true either of the whole of the subject or of part of it, so the proposition as it stands is plainly incomplete. But if we try to fix this by supplying the marks of quantity ourselves, we step outside the proper boundaries of logic, because we are assuming that we know the subject matter, or the science, that the proposition is about. We can safely take the examples above to mean "some metals are useful" and "all comets are subject to the law of gravitation," but not on logical grounds. So we may strike the class of indefinite propositions out of logic altogether, on the understanding that they must be made definite before we deal with them. I should point out, though, that in the lessons that follow I shall often use propositions in the indefinite form as examples, on the understanding that where no sign of quantity appears, the universal quantity is to be assumed. Probably, wherever a term is used on its own, it ought to be taken to mean the whole of its class. But however that may be, we need not recognize the indefinite proposition as a separate kind. And since singular propositions have been resolved into universals, only two kinds remain: Universal and Particular.

Remembering now that there are two kinds of proposition as regards quality, and two as regards quantity, we can form four varieties altogether, like this:

The vowels placed on the right are symbols, or shorthand names, that are always used for the four kinds of proposition. Their meaning is easy to remember once we notice that A and I occur in the Latin verb affirmo, I affirm, and E and O in nego, I deny.

There will usually be no difficulty in putting a proposition we meet in our reading into its proper class. The mark of universality is usually some adjective of quantity, such as all, every, each, any, none; but whenever the predicate is clearly meant to apply to the whole of the subject, we may treat the proposition as universal. The signs of a particular proposition are the adjectives of quantity some, certain, a few, many, most, or any others that clearly indicate part at least.

A negative proposition is recognized by the adverbial particle not joined to the copula. But in the proposition E, that is, the universal negative, we often put the particle no or none in front of the subject instead. So "no metals are compound" and "none of the ancients were acquainted with the laws of motion" are familiar forms of the universal negative.

The student must also always be ready to meet misleading or ambiguous ways of putting things. The proposition "all the metals are not denser than water," for instance, might be taken as E or as O, depending on whether we read it as "no metals are denser than water" or as "not all the metals are denser than water," and the second, of course, is the true sense. The little adjective few is especially liable to a subtle ambiguity of this kind. If I say "few books are at once learned and amusing," I may fairly be taken to assert that a few books certainly are both; but what I really mean to draw attention to is my belief that "the greater number of books are not at once learned and amusing." A proposition of this kind should generally be classed as O rather than I. The word some suffers from exactly the same sort of ambiguity: it can mean "some but not all," or "some at least, and perhaps all." The second appears to be the correct interpretation, as the next lesson shows (see the table in Lesson 9 of how the truth of each proposition bears on the others).

As we meet propositions in ordinary language, they come with various inversions and other departures from the simple logical form.

(1) It is quite common, especially in poetry, to find the predicate placed first, for emphasis or variety, as in "Blessed are the merciful," "Comes something down with eventide," "Great is Diana of the Ephesians." Such an inversion of the terms is usually easy to spot, and the sentence must then be put back into the regular order before logic can deal with it.

(2) The subject can sometimes be mistaken for the predicate when it is described by a relative clause standing at the end of the sentence, as in "no one is free who is enslaved by his appetites." Here free is clearly the predicate, although it stands in the middle of the sentence, and "one who is enslaved by his appetites" is the real subject. This proposition is clearly of the form E.

Propositions are also expressed in various ways that differ from the simple logical order, and we must notice some of the kinds that result.

Exclusive propositions contain words such as only, alone, none but, which limit the predicate to the subject. In "elements alone are metals," we are told that the predicate "metal" cannot be applied to anything except "elements," but we are not to understand that "all elements are metals." The same meaning is expressed by "none but elements are metals," or again by "all that are not elements are not metals," and we shall see in the next lesson that this is really equivalent to "all metals are elements." Arguments that look fallacious at first sight will often turn out to be correct when they contain exclusive propositions and these are properly interpreted.

Exceptive propositions affirm a predicate of all the subject except certain stated cases, to which, by implication, the predicate does not belong. "All the planets, except Venus and Mercury, are beyond the earth's orbit" is clearly a proposition equivalent to two: that Venus and Mercury are not beyond the earth's orbit, and that the rest are. If the exceptions are not actually named, an exceptive proposition must often be treated as a particular one. Suppose I say "all the planets in our system except one agree with Bode's law" and do not name that one exception. Then the reader cannot, on the strength of the proposition, positively assert of any planet that it does agree with Bode's law.

Some propositions are distinguished as explicative, or essential, because they simply affirm of their subject a predicate that everyone who can define the subject knows belongs to it. Such propositions only unfold what is already contained in the subject. "A parallelogram has four sides and four angles" is an explicative, or essential, proposition. "London, which is the capital of England, is the largest city of Europe" contains two propositions, and one of them only directs our attention to a fact everyone may be supposed to know, namely that London is the capital of England.

Ampliative propositions, on the other hand, join a new predicate to the subject. So, for anyone who does not know how the cities of Europe compare in size, the last example contains an ampliative proposition. Most propositions are of this kind.

Tautologous, or Truistic, propositions are those that simply affirm the subject of itself and give no information whatever, as in "whatever is, is," or "what I have written, I have written."

It is no part of formal logic to teach us how to interpret the meanings of sentences as we meet them in our reading; that is more the work of the grammarian and the philologist. Logic deals with the relations between different propositions and the inferences that can be drawn from them. Even so, it is a good idea for the reader to get some familiarity with the real logical meaning of conventional or peculiar forms of expression, and a number of examples will be found in the Examples of Propositions at the end of Lesson 10, which the reader is asked to classify and treat as directed.

Besides the distinctions already noticed, it has long been usual to distinguish propositions as pure or modal. The pure proposition simply asserts that the predicate does or does not belong to the subject. The modal proposition asserts this cum modo, that is, "with a mode": with some indication of the mode or manner in which the predicate belongs to the subject. Any adverb of time, place, manner, degree and so on, or any expression equivalent to an adverb, makes a proposition modal. "Error is always in haste," "justice is ever equal," "a perfect man ought always to be conquering himself" are examples of modal propositions in this sense of the name. Other logicians, however, have taken a different view, and treat modality as the degree of certainty or probability with which a judgment is made and asserted. So we may say "an equilateral triangle is necessarily equiangular," "men are generally trustworthy," "a falling barometer probably indicates a coming storm," "Aristotle's lost treatises may possibly be recovered," and each of these assertions is made with a different degree of certainty, or modality. Dr. Thomson is no doubt right in holding that modality does not affect the copula of the proposition, and that the subject could only be properly dealt with in a work on Probable Reasoning.

Many logicians have also divided propositions into true and false, and this might well seem an important distinction. All the same, it lies wholly outside the logician's province to consider whether a proposition is true or not in itself. All the logician has to determine is the comparative truth of propositions, that is, whether one proposition is true when another is. Strictly speaking, logic has nothing to do with a proposition on its own. The work of reasoning consists only in converting or transforming certain propositions into certain others, and the truth of the conclusion is in question only insofar as it follows from the truth of what we shall call the premises. It is the duty of each special science, in its own sphere, to decide which propositions are true and which are false. If logic took that duty on itself, it would be just another name for the whole of knowledge.

See Mr. J. S. Mill's System of Logic, Book I, Chap. iv, which generally agrees with what is given above. Chapters V and VI contain Mill's views on the Nature and Import of Propositions, a subject that can be studied further in Mill's Examination of Sir William Hamilton's Philosophy, Chap. xviii; Hamilton's Lectures on Logic, No. XIII; and Mansel's Prolegomena Logica, Chap. II. But the subject is too metaphysical in character to be dealt with in this book.

Questions and Exercises

1. Define a proposition, and name the parts it is made of.

2. How are propositions classified?

3. Name the four kinds of categorical propositions, and their symbols.

4. Under which classes are singular and indefinite propositions placed?

5. List the most usual signs of the quantity of a proposition.

6. What are modal propositions according to the early logicians, and according to Thomson?

7. How far do logicians consider propositions with regard to their truth or falsity?

Lesson 9: The Opposition of Propositions

We have found that logicians recognize four distinct kinds of propositions: the Universal affirmative, the Particular affirmative, the Universal negative, and the Particular negative, usually marked by the symbols A, I, E, O. Now we need to compare these four kinds fairly closely, looking at what each one means and how each is used. That way we will learn clearly how the truth of one affects the truth of the others, and how the same truth can be put into different forms of words.

The proposition A expresses the fact that the thing, or class of things, denoted by the subject is included in the class of things denoted by the predicate, and forms part of it. So "all metals are elements" means that the metals form a part of the class of elements, but not the whole of it. In the 1870s, 63 elements were known altogether, and 48 of them were metals, so we cannot say that all elements are metals. In fact the proposition does not tell us anything about elements in general. It is not really about elements at all: metals are the subject, and it is the metals it gives us information about. The best way to show this is with a kind of diagram first used by the famous mathematician Euler, in his letters to a German princess. In Fig. 1, imagine the metals shut inside the small circle, rather as sheep might be shut in a pen; this circle holds all the metals and nothing else. Imagine the larger circle holding, in the same way, all the elements and nothing else. Since the small circle lies wholly inside the larger one, all the metals must be counted as elements. But about the part of the elements lying outside the circle of metals, the proposition tells us nothing.

The particular affirmative proposition I means exactly the same kind of thing as A, except that only part of the subject is brought into question. When I say that "some metals are brittle," I mean that if we gathered together all the different metals, at least a few could be picked out that would turn out to be brittle. But the word some is extremely vague. It shows neither exactly how many metals are brittle, nor how we are to tell them from the others, except, of course, by testing whether they are brittle. In Euler's way, this proposition is properly shown by two intersecting circles, one supposed to enclose all metals and the other all brittle substances. The mere fact that the two circles intersect proves that some part of one class must coincide with some part of the other, which is just what the proposition means to express. About the parts of the circles that do not overlap, the proposition tells us nothing.

The universal negative proposition E denies that there is any agreement or coincidence at all between the subject and the predicate. From "no metals are compound substances," for example, we learn that no metal is to be found among compound substances. And it follows necessarily that no compound substance can be found among the metals. For if there were a compound substance among the metals, there would plainly be at least one metal among the compound substances. Euler's method shows this complete separation of the two classes in thought very well, by two circles that do not touch.

The reader will easily see what sets the proposition E apart from A and I. E gives us some information about the whole of the predicate, because we learn that none of the things included in the predicate can be found among the things included in the subject. The affirmative propositions, on the other hand, entitled us to hold that the things denoted by the subject, or some particular part of them, were included in the predicate. But they give us no right to say that any particular part of the predicate is in the subject. If all we know about a substance is that it is an element, the proposition "all metals are elements" does not tell us whether it is a metal or not. And from the proposition "some metals are brittle" we certainly cannot find out whether any particular brittle substance is a metal. We have to get that information from somewhere else. But from "no metals are compounds" we learn, of any compound substance, that it is not a metal, and also, of any metal, that it is not a compound substance.

In technical language, this important difference is expressed by saying that the proposition E distributes its predicate, while the affirmative propositions A and I do not distribute their predicates. The distribution of a term simply means taking it universally, or referring to all parts of it. Whether an argument or syllogism is valid will usually depend on whether its terms are sufficiently distributed, so we cannot pay too much attention to this point.

Judging from the examples we have had, we can see that the universal affirmative distributes its subject, but not its predicate: it gives us some information about all metals, but not about all elements. The particular affirmative distributes neither subject nor predicate: our example teaches us nothing about all metals, or about all brittle substances. But the universal negative distributes both subject and predicate, because we learn something about all metals and also about all compound substances.

The particular negative proposition O turns out to distribute its predicate, but not its subject. When I say that some metals are not brittle, I deliberately refer to only part of the metals, and I shut them out of the class of brittle substances. But in doing so I cannot help referring at the same time to the whole of the brittle substances. If the metals in question coincided with any part at all of the brittle substances, they could not be said to be shut out of that class. To shut a thing out of any space, such as one particular room of a house, it is not enough to remove it from some part of the room; it must be kept out of every part, out of the whole of that space or room. Euler's diagram for this proposition can be drawn in the same way as the one for the proposition I, like this:

Here, although part of the metals falls inside the circle of brittle substances, the rest of the metals are shut out of every part of the predicate.

We can set out the result we have now reached in the following form:

Now we can very easily work out how the four propositions are related to one another, each to each; that is, the ways in which they are opposed to each other. Clearly, the truth of one proposition may interfere, more or less completely, with the truth of another proposition that has the same subject and predicate. If "all metals are elements," then it is impossible that "some metals are not elements," and still more obviously impossible, so to speak, that "no metals are elements." So the proposition A is inconsistent with both E and O, and, the other way around, E and O are inconsistent with A. In the same way, E is inconsistent with A and I. But there is an important difference to notice here. If A is false, O is necessarily true, but E may or may not be true. If it is not true that "all men are sincere," it follows that "some men are not sincere," but it does not in the least follow that "no men are sincere." We express this difference by saying that A and O are contradictory propositions, while A and E are called contrary propositions. A and E, as in "all men are sincere" and "no men are sincere," plainly stand for the greatest possible opposition of circumstances. To prove that A is false, it is enough to establish that O is true; it is unnecessary to prove E, even if that could be done. In the same way, E is disproved by proving I, and it is unnecessary to prove A. Anyone who asserts a universal proposition, either A or E, takes on the burden of explaining away or disproving every single exception that is brought against it. An opponent can always stick to the much easier job of finding cases that seem to contradict, or really do contradict, the universality of the statement. But if he takes it on himself to assert the direct contrary, he is the one open to easy attack. Suppose someone asserted, for instance, that "All Christians are more moral than Pagans." It would be easy to bring forward examples showing that "Some Christians are not more moral than Pagans." But it would be absurd to think it necessary to go to the opposite extreme and show that "No Christians are more moral than Pagans." In short, A is sufficiently, and best, disproved by O, and E by I. And it is easy to see that, the other way around, O is disproved by A, and I by E. In fact, there is no other way at all of disproving these particular propositions.

When we compare the propositions I and O, we find that they are opposed in nature in a certain sense, since one is affirmative and the other negative, and yet they are consistent with each other. It is true, for instance, that "Some metals are brittle": antimony, bismuth and arsenic, for example. But it is also true that "Some metals are not brittle." And the reader will notice that when I assert "Some metals are elements," nothing in this prevents "Some metals are not elements" from being true as well, although we know on other grounds that it is not true. The propositions I and O are each called the subcontrary of the other. The name implies a lesser degree of contrariety than there is between A and E.

As for the relation of A to I, and of E to O, the truth of the universal plainly includes and requires the truth of the particular. Whatever can be affirmed or denied of all parts of a class can certainly be affirmed or denied in the same way of some part of the class. But from the truth of the particular we have no right to infer either the truth or the falsity of the universal of the same quality. These pairs of propositions are called subalterns, meaning "one under the other" (from the Latin sub, under, and alter, the other of two). Or, to put it more exactly, I and O are the subalternates of A and E respectively, and each of A and E is a subalternans.

The relations of the propositions we have just described are all shown clearly in the following scheme:

It is so very important to grasp the consistency or opposition of propositions completely and quickly that I will put the matter in another form. If we take any two propositions with the same subject and predicate, they must come under one of the following statements:

1. Of contradictory propositions, one must be true and one false.

2. Of contrary propositions, both cannot be true, and both may be false.

3. Of subcontrary propositions, only one can be false, and both may be true.

4. Of subalterns, the particular is true if the universal is true; but the universal may or may not be true when the particular is true.

I put the same matter in yet another form in the following table, which shows how the truth of each of A, E, I and O affects the truth of each of the others.

Clearly, we get more information from affirming universals than from affirming particulars. It follows that we can get more information from denying particulars than from denying universals: that is, fewer cases are left doubtful, as the table above shows.

The reader should be warned, though, against an ambiguity that has misled even some of the most eminent logicians. In particular propositions, the adjective some must be carefully understood as meaning some, and there may or may not be more, or all. If we took it to mean some, but not more and not all, it would really give the proposition the force of I and O combined. If I say "some men are sincere," I must not be taken to imply that "some men are not sincere." I must be understood to predicate sincerity of some men, and to leave the character of the rest entirely untouched. It follows that when I deny the truth of a particular, I must not be understood to imply the truth of the universal of the same quality. It might seem very natural to deny the truth of "some men are mortal," on the ground that not just some but all men are mortal. But then the proposition being denied would really be "some men are not mortal," that is, O, not I. So when I deny that "some men are immortal," I mean that "no men are immortal"; and when I deny that "some men are not mortal," I mean that "all men are mortal."

It has long been the custom to compare propositions according to the kind of subject matter they refer to. What is technically called the matter was divided into three kinds: necessary, contingent, and impossible. Necessary matter is any subject in which the proposition A can be affirmed; impossible matter is any subject in which E can be affirmed. Any subject or branch of knowledge in which universal statements usually cannot be made is called contingent matter, and it implies the truth of I and O. So "comets are subject to gravitation," although it is an indefinite or indesignate proposition (see Lesson 8), may be read as A, because it belongs to a part of natural science where general laws of that kind hold. But "men are sincere" would properly be read as particular, or I, because the matter is clearly contingent. The truth of the following statements is evident.

In necessary matter A and I are true; E and O false.

In contingent matter I and O are true; A and E false.

In impossible matter E and O are true; A and I false.

In reality, though, this part of logical doctrine is thoroughly illogical. As we have already explained (in Lesson 8), when we deal with a proposition we have no right to assume that we know the science it refers to. Our job is to draw out the exact consequences of whatever statements we are given. In logic we must learn to transform information in every possible way, but not to add facts from outside.

Questions and Exercises

1. State the quantity of the subject and of the predicate in each of the propositions A, E, I, O.

2. From the following propositions, pick out pairs of contrary, contradictory, subaltern, and subcontrary propositions:

(1) Some elements are known.

(2) No elements are known.

(3) All elements are known.

(4) Not all elements are known.

(5) Some elements are not known.

(6) All elements are not known.

3. What propositions are true, false, or doubtful:

(1) when A is false,

(2) when E is false,

(3) when I is false,

(4) when O is false?

4. Prove, by means of the contradictory propositions, that subcontrary propositions cannot both be false.

5. Show, by means of the subcontrary propositions, that contrary propositions may both be false.

6. What quantity would you assign to each of the following propositions?

(1) Knowledge is power.

(2) Nebulae are material bodies.

(3) Light is the vibration of an ether.

(4) Men are more to be trusted than we think.

(5) The Chinese are industrious.

7. Why is it better, in an argument, to refute a statement by its contradictory rather than by its contrary?

Lesson 10: Conversion of Propositions, and Immediate Inference

We infer whenever we draw one truth from another truth, or pass from one proposition to another. As Sir William Hamilton puts it, Inference is "the carrying out into the last proposition what was virtually contained in the antecedent judgments." Teaching the principles on which this act of inference must be carried out is the true business of logic. Everything we have said so far about terms and propositions is useful, or even relevant, only so far as it helps us understand how inference works. So we now have to look, one after another, at all the ways in which the same information can be recast into different forms of words, forms that often seem to say quite different things. Logicians do not agree exactly about what should be counted as inference and what should not. Everyone would agree that we make an inference when we see drops of water on the ground and believe that it has rained. That is a fairly complicated act of inference, and we will come to it in later lessons, under the heading of Induction. Few people, or none, would say that we make an inference when we pass from "The Duke of Cambridge is the Commander-in-chief" to "The Commander-in-chief is the Duke of Cambridge." (The Duke of Cambridge was then the head of the British Army.) But without worrying too much about what to call the process, I will point out in this lesson every way in which we can pass from a single proposition of the form A, E, I or O to another proposition.

We are said to convert a proposition when we swap its subject and predicate around. The proposition we get is called the converse, or converted proposition, and the one we started from is called the convertend, meaning the one to be converted. For the converse to be validly inferred from the convertend, we must keep two rules: (1) the quality of the proposition, affirmative or negative, must stay the same; and (2) no term may be distributed in the converse unless it was distributed in the convertend.

Suppose that in "all metals are elements" we simply swapped the terms, giving "all elements are metals." That would claim some knowledge about all elements. But we have already seen clearly that the predicate of A is undistributed, so the convertend really tells us nothing about all elements. All we can infer is that "some elements are metals." This converse obeys the rule, and the process by which we pass in this way from A to I is called Conversion by Limitation, or per accidens.

When the converse has exactly the same form as the convertend, the process is called simple conversion. From "some metals are brittle substances" I can infer "some brittle substances are metals," because every term here is undistributed. So I is simply converted into I.

Again, from "no metals are compounds" I can pass straight to "no compounds are metals." Both of these propositions are E propositions, so all their terms are distributed. Euler's diagram in Lesson 9 (Figure 3) shows clearly that if all the metals are kept apart from all the compounds, then all the compounds are necessarily kept apart from all the metals. So the proposition E is simply converted into E.

When we try to convert the proposition O, though, we run into a peculiar difficulty. Its subject is undistributed, and yet conversion would make that subject the predicate of a negative proposition, and a negative proposition distributes its predicate. Take the proposition "some existing things are not material substances." Converted directly, it would become "all material substances are not existing things," which is plainly absurd. The fallacy comes from the fact that "existing things" is distributed in the converse, while it was particular in the convertend; and the rules of Aristotle's logic do not allow us to put a sign of particular quantity in front of the predicate. The converse would be just as untrue and fallacious if we made the subject particular, as in "some material substances are not existing things." So we must conclude that the proposition O cannot be dealt with either by simple conversion or by conversion by limitation. We need a new process, which we may call Conversion by Negation. It consists in first changing the convertend into an affirmative proposition, and then converting it simply. If we attach the negation to the predicate instead of to the copula, our proposition becomes "some existing things are immaterial substances." Converting this simply, we get "some immaterial substances are existing things," and this can truly be inferred from the convertend. So the proposition O can be converted only by this exceptional method, by negation.

There is another process of conversion that can be applied to the proposition A, known as conversion by contraposition. From "all metals are elements" it necessarily follows that "all not-elements are not metals." If this is not obvious at first, a little thought will make it so. Figure 5 shows that if all the metals are among the elements, then whatever is not an element, that is, whatever lies outside the circle of elements, must also lie outside the circle of metals. If we may look ahead to what Lesson 23 contains, we can also prove that the contrapositive proposition is true in this way. Suppose something that is not-element were a metal. Then by the original proposition it would have to be an element, so it would be an element and not an element at once. That is impossible by the Primary Laws of Thought (Lesson 14), since nothing can both have and not have the same property. It follows that whatever is not-element must be not-metal.

It is easy to make mistakes in contrapositive conversion, for a reason that will become clearer in Lesson 22. We are very apt to infer from a proposition of the form "all metals are elements" that all not-metals are not-elements. That is not only a false statement in itself; it is not in the least justified by the original proposition. Figure 5 makes it plain that a thing which lies outside the circle of metals does not necessarily lie outside the circle of elements, which is wider than the circle of metals. All the same, people make this mistake often in everyday life. The reader will do well to remember that conversion by contraposition consists only in this: we take the negative of the predicate of the proposition A as a new subject, and we affirm of it, universally, the negative of the old subject.

Contrapositive conversion cannot be applied at all to the particular propositions I and O, nor to the proposition E as it stands. But we can change E into A by attaching the negation to the predicate, and then the process can be applied. So "no humans are perfect" can be changed into "all humans are not-perfect," that is, "are imperfect." Then by contraposition we infer "all not-imperfect beings are not-humans." But not-imperfect really means the same as perfect, so our new proposition really amounts to "all perfect beings are not humans," or "no perfect beings are humans" (E), which is the simple converse of the original proposition.

That leaves certain deductions that can be drawn from a proposition without converting its terms. We may call them immediate inferences, and Archbishop Thomson has described them very clearly in his "Outline of the Necessary Laws of Thought" (pp. 156, &c.).

Immediate Inference by Privative Conception consists in passing from any affirmative proposition to a negative proposition that is implied in it or equivalent to it, or the other way round, passing from a negative proposition to the affirmative that corresponds to it. It is also called Obversion.

The following table takes a proposition of each kind and changes it, by privative conception, into an equivalent proposition:

The truth of any of these can be clearly shown by diagrams. For instance, it will be plain that if the whole circle of metals lies inside the circle of elements, no part of it can lie outside that circle, among the compounds. Any of the propositions above can be converted, but the results will mostly be ones we have already obtained. The simple converse of "no metals are compounds" is "no compounds are metals," which is to say "no not-elements are metals," the contrapositive of "all metals are elements." From the last example in the table we also get, by simple conversion, "some untrustworthy beings are humans," which is obviously the converse by negation explained above. Applying this kind of conversion to "some humans are not untrustworthy," we get "some not-untrustworthy beings are humans." Finally, from "all humans are imperfect" we can obtain, by conversion by limitation, "some imperfect beings are humans."

Immediate Inference by Added Determinants consists in attaching an adjective, or some similar qualification, to both the subject and the predicate of a proposition, so that the meaning of each term becomes narrower, or more fully determined. As long as nothing else is altered, the truth of the new proposition follows necessarily from the truth of the original in almost every case.

So from "all metals are elements" we may infer that "all very heavy metals are very heavy elements." From "a comet is a material body" we infer "a visible comet is a visible material body." But if we use this kind of inference too boldly, we can arrive at fallacious and absurd results. From "all kings are men" we might infer "all incompetent kings are incompetent men." But it does not follow at all that someone who is incompetent as a king would be incompetent in other positions. In this case and many others, the qualifying adjective is apt to carry different meanings in the subject and in the predicate, and the inference is necessarily true only when the meaning is exactly the same in both. This kind of inference will seldom work with comparative terms. From "a cottage is a building," for example, we cannot infer "a huge cottage is a huge building," since a cottage may be large compared with other cottages but not compared with buildings in general.

Immediate Inference by Complex Conception is very like the last kind. It consists in using the subject and predicate of a proposition as parts of a more complex conception. From "all metals are elements" I can pass to "a mixture of metals is a mixture of elements." From "a horse is a quadruped" I infer "the skeleton of a horse is the skeleton of a quadruped." But here again the reader must beware of applying the process where the new complex conception means something different in the subject and in the predicate. From "all Protestants are Christians," it does not follow that "a majority of Protestants are a majority of Christians," nor that "the most excellent of the Protestants is the most excellent of the Christians."

I recommend that the student get thoroughly familiar with all the transformations of propositions, or immediate inferences, described in this lesson, and plenty of examples are given for the purpose. A good exercise is to put the same proposition through a series of changes so that it comes back to its original form at the end, which proves that all the changes along the way were sound. But if conversion by limitation has been used anywhere in the chain, the original universal proposition cannot be recovered, only the particular proposition that corresponds to it.

On Immediate Inference, see Archbishop Thomson, Outline of the Laws of Thought, §§ 85—92.

Questions and Exercises

1. Define inference and conversion.

2. What are converse and convertend propositions?

3. State the rules of valid conversion.

4. Name all the kinds of conversion.

5. By what process do we pass from each of the following propositions to the next?

(1) No knowledge is useless.

(2) No useless thing is knowledge.

(3) All knowledge is not-useless.

(4) All knowledge is useful.

(5) What is not useful is not knowledge.

(6) What is useless is not knowledge.

(7) No knowledge is useless.

6. Give the logical opposites of the following proposition, and the converse of its contradictory: "He cannot become rich who will not labor."

7. Apply negative conception to the proposition "All humans are fallible"; then convert it, and show that the result is the contrapositive of the original.

8. Sort the propositions below into the following four groups (an expedient act is one that is useful or advantageous): a. Those that can be inferred from (1). b. Those from which (1) can be inferred. c. Those that do not contradict (1), but cannot be inferred from it. d. Those that contradict (1).

(1) All just acts are expedient acts.

(2) No expedient acts are unjust.

(3) No just acts are inexpedient.

(4) All inexpedient acts are unjust.

(5) Some unjust acts are inexpedient.

(6) No expedient acts are just.

(7) Some inexpedient acts are unjust.

(8) All expedient acts are just.

(9) No inexpedient acts are just.

(10) All unjust acts are inexpedient.

(11) Some inexpedient acts are just acts.

(12) Some expedient acts are just.

(13) Some just acts are expedient.

(14) Some unjust acts are expedient.

Examples of Propositions

The reader should work out the logical character of each of the following propositions. For each one, say whether it is affirmative or negative; universal, particular, singular or indefinite; pure or modal; exclusive or exceptive; and so on. Where a proposition is stated irregularly, put it into the simple logical order. Then convert the proposition, and draw immediate inferences from it by any process that applies.

(1) All birds are feathered.

(2) No reptiles are feathered.

(3) Fixed stars are self-luminous.

(4) Perfect happiness is impossible.

(5) Life everyone holds dear.

(6) Every mistake is not a proof of ignorance.

(7) Some of the most valuable books are seldom read.

(8) He jests at scars who never felt a wound.

(9) Heated metals are softened.

(10) Not one of the Greeks at Thermopylae escaped.

(11) Few are acquainted with themselves.

(12) Whoever loves instruction loves knowledge.

(13) Nothing is harmless that is mistaken for a virtue.

(14) Some of our muscles act without volition.

(15) Metals are all good conductors of heat.

(16) Fame is no plant that grows on mortal soil.

(17) Only the brave deserve the fair.

(18) No one is free who does not command himself.

(19) Nothing is beautiful except truth.

(20) The wicked shall fall by his own wickedness.

(21) Unsafe are all things unbecoming.

(22) There is no excellent beauty that has not some strangeness in the proportion.

(23) It is a poor center of a man's actions, himself.

(24) Mercy but murders, pardoning those that kill.

(25) I shall not all die. (Non omnis moriar.)

(26) A regiment consists of two battalions.

(27) It is cruelty to load a falling man.

(28) Every mistake is not culpable.

(29) Quadrupeds are vertebrate animals.

(30) Not many of the metals are brittle.

(31) Many are the deserving people who are unfortunate.

(32) Amalgams are alloys of mercury.

(33) One kind of metal at least is liquid.

(34) Talents are often misused.

(35) Some parallelograms have their adjoining sides equal.

(36) Britain is an island.

(37) Romulus and Remus were twins.

(38) A man's a man.

(39) Heaven is all mercy.

(40) Everyone is a good judge of his own interests.

(41) All parallelograms have their opposite angles equal.

(42) Familiarity breeds contempt.

(43) No one is always happy.

(44) Many a little makes a mickle (a great deal).

Lesson 11: Logical Analysis of Sentences

The propositions we meet in ordinary writing and speech seldom show the simple form we have seen to be the proper logical construction: a subject, a copula and a predicate joined together. Often the copula is mixed up with the predicate. Often, too, several propositions are packed into one grammatical sentence. For a full account of how sentences are analyzed, I will point the reader to several excellent little books on the subject. Here I will only try to sketch the different ways a sentence can be built.

Footnote: W. S. Dalgleish's Grammatical Analysis, or J. D. Morell's Analysis of Sentences. Alex. Bain's English Composition and Rhetoric, pp. 91–117, deals with the construction of sentences.

In ordinary language the copula is so often joined to the predicate that grammarians treat a proposition as having only two parts: the subject, and the predicate, or verb. Take the proposition "The sun rises." It seems to contain nothing but a subject, "the sun," and a predicate, "rises." But it really means the same as "the sun is rising," where the copula is plainly shown. So we will treat the verb, or grammatical predicate, as containing both the copula and the logical predicate. In Latin a single word can hold all three parts of a proposition, as in sum, "I am." Caesar's famous exclamation Veni, vidi, vici, "I came, I saw, I conquered," holds three separate and complete propositions in three words. But these odd cases arise only because the parts of the proposition have been blended together and disguised inside one word. In the Latin sum, the letter m is a leftover of the pronoun me, which is the real subject of the proposition. If we knew the grammar of any language perfectly, it would probably not contradict the logical view of a sentence. Instead it would probably explain how the separate parts of the complete proposition got blended together and seemingly lost, just as the words will and not are blended in the everyday "I won't."

A grammatical sentence can contain any number of separate propositions. They could be pulled apart, but they are combined to save words. In the sentence

	"Art is long and Time is fleeting,"

there are two separate subjects, Art and Time, and two predicates, "long" and "fleeting." So we simply have two propositions joined by the conjunction and. But we can also have several separate subjects with one and the same predicate, as in

	"Thirty days hath September,
	April, June, and November."

In this well-known couplet the predicate, "having thirty days," is put first for emphasis, and there are four subjects, September, April and the rest, and the predicate is affirmed of each of them. So these lines really contain four separate propositions.

Or there can be one subject with several predicates, so that several different propositions are asserted without repeating the subject and the copula. The sentence

"Nitrogen is a colourless, tasteless, inodorous gas, slightly lighter than air," has only one subject, Nitrogen, but four or five predicates ("inodorous" means odorless). It plainly means the same as "Nitrogen is colourless," "Nitrogen is tasteless," "Nitrogen is a gas," and so on.

Last, we can have several subjects and several predicates all combined in one sentence with only one copula, so that each predicate is asserted of each subject. Then a great many separate propositions are squeezed into one short sentence. In the sentence "Iron, Copper, Lead and Zinc are abundant, cheap and useful metals," we clearly have four subjects, and we can say we have four predicates: "abundant," "cheap," "useful" and "metal." Nothing stops us from applying each predicate to each subject, so the sentence really contains 16 separate propositions in only 11 words: "Iron is abundant," "Iron is cheap," "Copper is abundant," "Copper is cheap," and so on. In this curious sentence,

"Hearts, tongues, figures, scribes, bards, poets, cannot think, speak, cast, write, sing, number, his love to Antony," Shakespeare has joined six subjects and six predicates, or verbs, so that, strictly speaking, there are six times six, or thirty-six, propositions.

Footnote: Antony and Cleopatra, Act III. Sc. i.

In all the cases above, the sentence is said to be compound, and the separate propositions combined in it are said to be coordinate with each other, that is, of the same order or rank. They are called that because they do not depend on each other or affect each other's truth in any way. There is no need to state how abundant, cheap or useful iron is in the same sentence as the qualities of copper, lead or zinc. But since the predicates happen to be the same, we save a good deal of trouble in speaking or writing by attaching as many subjects as we can to the same set of predicates. It is truly said that brevity is the soul of wit, and one of the great arts of writing is to pack as many statements as possible into the fewest words, as long as the meaning does not get muddled.

Propositions are combined in a completely different way when one proposition forms part of the subject or the predicate of another. In the sentence "The man who is upright need not fear accusation," there are two verbs and two propositions, but one of them only describes the subject of the other. "Who is upright" clearly limits the predicate "need not fear accusation" to one part of the class "man." The meaning of the whole sentence could be put like this:

	"The upright man need not fear accusation."

Here we can clearly see that the clause, or apparent proposition, has taken the place of an adjective. A clause or proposition like this is called subordinate, because it only helps to build the principal sentence and has no meaning apart from it. Any sentence that contains a subordinate clause is said to be complex. Almost any part of a sentence can be replaced by a subordinate clause in this way. In "Oxygen and Nitrogen are the gases which form the largest part of the atmosphere," a subordinate clause makes up part of the predicate, and the meaning could be expressed almost as well like this: "Oxygen and Nitrogen are the gases forming the largest part of the atmosphere."

Take a modal proposition (see Lesson 8), one that states the manner in which the predicate belongs to the subject. Its mode can be expressed either by an adverb or by a subordinate clause. "As a man lives so he dies" is a proposition of this kind. It means "a man dies as he lives," and "as he lives" does the job of an adverb: if he lives well, he dies well; if he lives badly, he dies badly. Adverbs or adverbial clauses can also specify the time, the place, or any other circumstance the truth of the main proposition depends on.

Assuming the reader knows the grammatical terms involved, we can now list the parts that even the most complex sentence must be made of. The subject may be made up of:

1. A noun, as in "The Queen reigns."

2. A pronoun, as in "She reigns."

3. An adjective turned into a noun, as in "The brave are honored."

4. A gerund, as in "Seeing is believing."

5. An infinitive, as in "To see is to believe."

6. A subordinate clause, as in "Who falls from virtue is lost."

The subject may be qualified or restricted by joining an attribute to it, and the attribute can be expressed in any of these ways:

1. An adjective, as in "Fresh air is wholesome."

2. A participle, as in "Falling stars are often seen."

3. A noun used as an adjective, as in "Iron ships are now much employed."

4. A noun and a preposition, as in "ships of iron are now much employed."

5. A possessive case, as in "Chatham's son was the great minister Pitt."

6. A noun in apposition, as in "The Metropolis London is the most populous of cities."

7. A gerund or a dative infinitive (an infinitive saying what something is for), as in "The desire to go abroad is common in Englishmen."

The predicate is almost always made up of a verb, which often has an object or some qualifying words. So it may be:

1. A simple tense of a complete verb, as in "The sun rises."

2. A compound tense, as in "The sun has risen."

3. An incomplete verb and its complement, as in "The sea seems rough."

4. The verb "to be" and an adjective, as in "Time is fleeting."

5. A verb with an object, as in "Warmth melts ice."

6. A verb with an adverbial, as in "The snow falls thickly."

The object of a verb is usually a noun or a pronoun, but any of the six kinds of expression that can serve as a subject can also serve as an object.

The adverbial that qualifies a verb, and expresses the manner, time, place or some other circumstance affecting the proposition, may be:

1. An adverb, as in "The days pass slowly."

2. A noun and a preposition, as in "The resolution was passed by a large majority."

3. An absolute phrase, as in "The snow melts, the sun having risen."

4. A dative infinitive, as in "She stoops to conquer."

5. Any phrase that does the work of an adverb, as in "The dividends are paid twice a year."

People have invented various ways of showing how sentences are built, using symbols and names for the different parts. But I believe that by far the simplest and most effective way is to show the construction as a diagram. Any two or more parts of a sentence that are coordinate with each other, or that stand in the same relation to some other part, are written side by side and joined by a bracket. So the diagram

shows clearly that there are four coordinate subjects and four coordinate predicates in the example we took earlier.

Whenever one part of a sentence is subordinate to another part, we can connect the two with a line drawn in whatever direction is convenient. The analysis of the following sentence is easily shown by the diagram below it:

"No one who is a lover of money, a lover of pleasure, and a lover of glory, is likewise a lover of mankind; but only he who is a lover of virtue."

We see that the sentence is both compound and complex. That is, it contains two principal coordinate propositions with a shared predicate, "a lover of mankind." The first proposition is negative, and its subject is described by three subordinate clauses. The second proposition is affirmative and has one subordinate clause.

I will end this rather long lesson by analyzing a few sentences. The first is made up of some remarkably complex lines from a poem by Burbidge:

	"He who metes, as we should mete,
	Could we His insight use, shall most approve,
	Not that which fills most space in earthly eyes,
	But what—though Time scarce note it as he flies—
	Fills, like this little daisy at my feet,
	Its function best of diligence in love."

	"Most sweet it is with unuplifted eyes
	    To pace the ground, if path there be or none,
	While a fair region round the traveller lies
	    Which he forbears again to look upon;
	Pleased rather with some soft ideal scene,
	    The work of fancy, or some happy tone
	Of meditation slipping in between,
	    The beauty coming, and the beauty gone."
	                                        Wordsworth.

In this sentence there is clearly one subject, "to pace the ground," which is connected by the pronoun it to the predicate "most sweet." But the main part of the sentence is made up of three adverbials, which express the manner and the surrounding circumstances, and the third adverbial is developed in a very complicated way. The sentence is not compound, but it is complex, because of its four subordinate propositions.

In the next sentence there is, strictly speaking, only one principal proposition, "We find." But this is only a way of introducing what the sentence is really saying: "the two classes of intellectual operations have much that is different, much that is common."

"When the notions with which men are conversant in the common course of life, which give meaning to their familiar language and which give employment to their hourly thoughts, are compared with the ideas on which exact science is founded, we find, that the two classes of intellectual operations have much that is different, much that is common."

Here "the two classes" make a collective term, and they have two coordinate predicates, which so far makes the sentence a compound one. But most of the sentence is a complicated subordinate sentence that works like an adverbial: it expresses the time or occasion when this is found to be the case.

As a last example we take the sentence below:

"The law of gravitation, the most universal truth at which human reason has yet arrived, expresses not merely the general fact of the mutual attraction of all matter; not merely the vague statement that its influence decreases as the distance increases, but the exact numerical rate at which that decrease takes place; so that when its amount is known at any one distance it may be exactly calculated for any other."

Questions and Exercises

1. How does the grammatical predicate differ from the logical predicate?

2. Explain the difference between a compound sentence and a complex sentence, and between coordinate and subordinate propositions.

3. List the grammatical expressions that can form

(1) A subject.

(2) An attribute.

(3) A predicate.

(4) An object.

(5) An adverbial.

4. Examine the following sentences. Work out which are compound and which are complex, and point out the coordinate or subordinate propositions.

(1) Happy is the man that findeth wisdom, and the man that getteth understanding.

(2) Heat, being motion, can be converted into mechanical force.

(3) Ceres, Pallas, Juno, and Vesta are minor planets, or asteroids.

(4) Knowledge comes, but wisdom lingers.

(5) Fortune often sells to the hasty what she gives to those who wait.

(6) Thousands at His bidding speed, And post o'er land and ocean without rest; They also serve who only stand and wait.

(7) Pride that dines on vanity, sups on contempt.

(8) Nobody can be healthful without exercise, neither natural body, nor politic.

(9) Nature is often hidden, sometimes overcome, seldom extinguished.

(10) It is impossible to love and be wise.

(11) Though gods they were, as men they died.

(12) He that is not industrious envieth him that is.

(13) Ye are my friends, if ye do whatsoever I command you.—John 15:14.

(14) The wisdom that is from above is first pure, then peaceable, gentle, and easy to be intreated, full of mercy, and good fruits, without partiality, and without hypocrisy.—James 3:17.

5. Analyze any of the following sentences in the form of a scheme or diagram:

(1) The first aphorism of Bacon's Novum Organum, quoted in Lesson 27.

(2) Some judgments are merely explanatory of their subject, having for their predicate, a conception which it fairly implies, to all who know and can define its nature.

(3) There be none of the affections which have been noted to fascinate or bewitch, but love and envy: they both have vehement wishes; they frame themselves readily into imaginations and suggestions; and they come easily into the eye, especially upon the presence of the objects, which are the points that conduce to fascination, if any such there be.

For more examples to analyze, see Dalgleish's Grammatical Analysis with Progressive Exercises (Oliver and Boyd, Edinburgh, 1866; price 9d.).

Lesson 12: The Predicables, Division, and Definition

Before we go any further, the reader should get a precise grasp of a few logical terms known as the Predicables. These are the kinds of terms, or attributes, that can always be predicated of any subject. There are five of them: genus, species, difference, property, and accident. Used properly, they are extremely useful and important in logic. Ancient writers gave the predicables many different and subtle meanings, and it would be neither possible nor worthwhile to try to describe them all in this book. All we can give here is the simplest and most useful view of the subject.

Any class of things can be called a genus (from the Greek γένος, genos, race or kind) if we think of it as made up of two or more species. "Element" is a genus when we think of it as divided into the two species "metallic and non-metallic." Triangle is a genus with respect to the species acute-angled, right-angled, and obtuse-angled.

A species, on the other hand, is any class that we think of as forming part of the next larger class. So the terms genus and species are relative to each other. The genus is the larger class that gets divided, and the species are the two or more smaller classes it is divided into.

It is essential, though, to take these words in the two senses of extension and intension. From the explanation of those two senses in Lesson 5, it should be clear that the extent of a genus or species is simply the number of individuals it includes, and that there will always be fewer individuals in the species than in the genus. In extent, the genus book includes all books of every size, language, and subject. If we divide it by size, the species of book are folio, quarto, octavo, duodecimo, and so on (the old printers' names for book sizes, from the number of times the printed sheet was folded). Each of these species, of course, contains far fewer individual books than the whole genus.

In intension, the genus means not the individual things it contains but the sum of the qualities common to all those things, enough to mark them out clearly from other classes. In the same way, the species means the sum of the qualities common to all the individuals in that species, enough to mark them out from the rest of the genus as well as from everything else. So there must be more qualities implied in the meaning of the species than in the meaning of the genus. The species must contain all the qualities of the genus, plus some extra quality or qualities that tell the species apart from one another. These extra qualities form the difference. The difference can be defined as the quality, or sum of qualities, that marks out one part of a genus from the other part or parts. The difference (in Latin differentia, in Greek διαφορά, diaphora) has a meaning only in intension. When we use all these terms purely in intension, we can say that the difference added to the genus makes the species. If "building" is the genus, for example, and we add the difference "used for a dwelling," we get the species "house." If we take "triangle" as the genus, it means the sum of the qualities "three-sided rectilineal figure" (a figure bounded by straight lines). If we add the quality "having two sides equal," we get the species "isosceles triangle."

It is easy to see that the same class of things can be a genus and a species at the same time, depending on whether we think of it as divided into smaller classes or as part of a larger class. Triangle, which is a genus with respect to isosceles triangle, is a species with respect to right-lined geometrical figures. House is a species of building, but a genus with respect to mansion, cottage, villa, and the other kinds of houses. In fact we can have an almost endless chain of genera and species, each class being a species of the class just above it and a genus with respect to the class just below. Take the genus British subject. It has the species Born in the United Kingdom, Colonial-born, and Naturalized. Each of these becomes a genus with respect to the species male and female. Each of those species can be divided again into adult and minor; educated and uneducated; employed in some occupation or unemployed; self-supporting, supported by friends, or a pauper (living on public poor relief); and so on. The subdivision can go on like this until we reach a class so narrow in extent that it cannot be divided any further except into individuals. That class is called the lowest species, or infima species. All the genera and species in between are called subaltern (from the Latin sub, under, and alter, the other of two), because they stand one under another. If there is a genus that is not thought of as a species, that is, as part of any higher genus, it is called the summum genus, the highest genus, or genus generalissimum, the most general genus.

It is doubtful whether we can really put any such upper limit on the chain of classes. The class British subject is certainly not an absolute summum genus, since it is only a species of man, which is a species of animal, living being, part of the earth, substance, and so on. If there were any real summum genus, it would probably be "Being," or "Thing," or "Object that can be conceived." But we can usefully use the term to mean the highest class of things taken in by any science or classification. So "material substance" is the summum genus that the science of chemistry examines, and "inhabitant of the United Kingdom" is the summum genus that the British census counts and classifies. Logical terms are only a species of words or phrases, but they are the summum genus as far as logic is concerned, because logic has nothing to do with the parts of speech or with the relations of words, syllables, and letters that grammarians study.

Several very useful expressions come from the words genus and species. When a thing is so peculiar and so unlike other things that it can't easily be put into one class with them, we say it is sui generis, or "of its own genus." The rings of Saturn, for instance, are so different from anything else among the heavenly bodies that they can fairly be called sui generis. In zoology, the Ornithorhynchus, or Australian duck-bill (the platypus), the Amphioxus (the lancelet), and a few other animals are so peculiar that they can be called sui generis. When a substance is the same in all its parts, or when a number of things are all alike, we say they are homogeneous (from the Greek ὁμός, homos, like, and γένος, genos, kind), that is, of the same nature. Otherwise they can be called heterogeneous (from the Greek ἕτερος, heteros, other).

We must carefully separate the purely logical use of genus and species from their special use in natural history. There, a species is a class of plants or animals supposed to have descended from common parents and to be the narrowest class that has a fixed form, and the genus is the next class up. But if we accept Darwin's theory of the origin of species, this definition of species becomes entirely illusory, because according to that theory different genera and species must have descended from common parents. Species then means just an arbitrary amount of resemblance that naturalists choose to settle on, and that can't be defined any more exactly. So this use of the word has nothing whatever to do with the logical use, in which any class of things at all is a species, as long as we think of it as part of a wider class, or genus.

The fourth of the Predicables is Property (in Latin proprium, in Greek ἴδιον, idion, one's own). It is hard to define in a way that is free from objections and difficulties, but perhaps the best description is this: a property is any quality that is common to the whole of a class but is not needed to mark that class out from other classes. For example, it is a property of the genus "triangle" that its three inside angles add up to two right angles. This is a very remarkable fact, and it is always true of triangles. But it is not made part of the genus, and it is not used in defining a triangle, because having three straight sides is enough of a mark. Geometrical figures have a great many properties. The Second Book of Euclid is taken up with proving a few properties of rectangles, and the Third Book does the same for circles. As we usually use the word, a property may or may not belong to other things as well as the ones in question. Some properties of the circle may also belong to the ellipse, and some properties of human beings, such as the power of memory or of anger, may belong to other animals too.

Logicians have invented various subtle divisions of properties, but it is enough to say this. A peculiar property is one that belongs to the whole of a class and to that class only, as laughter is supposed to belong only to human beings; the property of enclosing the greatest area within a boundary of a given length is peculiar to circles. A property that is not peculiar may belong to other classes of things as well as to the class it is called a property of. We can also distinguish the Generic Property, which belongs to the whole of the genus, from the Specific Property, which belongs to the whole of a lowest species.

Last, an accident (in Latin accidens, in Greek συμβεβηκός, symbebekos) is any quality that may belong to a class or not, as the case may be, without affecting the class's other qualities. The word means what befalls or happens by chance, and has no necessary connection with the nature of a thing. The absolute size of a triangle, for example, is a pure accident as far as its geometrical properties go. Whether the side of a triangle is 1/10 of an inch or a million miles, whatever Euclid proves true of one is true of the other. A man's birthplace is an accident with respect to him, and so are the clothes he is wearing, the position he is resting in, and so on. Some writers distinguish separable and inseparable accidents. The clothes a man is wearing are a separable accident, because they can be changed, and so can his position and many other circumstances. But his birthplace, his height, his first name, and so on are inseparable accidents, because they can never be changed, even though they have no necessary or important connection with his general character.

To illustrate part of the scheme of classification described under the name of the Predicables, I will give here, as logic textbooks usually do, the Tree of Porphyry. It is a sort of model classification invented by one of the early Greek logicians, Porphyrius (Porphyry). I have simplified the usual form of it by translating the Latin names and leaving out unnecessary words.

In this Tree we see a succession of genera and species: Substance, Body, Living Being, Animal, and Man. Of these, Substance is the summum genus, because it is not treated as a species of any higher class. Man is the infima species, because it is a class that is not divided into any lower class, only into individuals, and the usual individuals to name are Socrates and Plato. Body, Living Being, and Animal are called subaltern genera and species, because each one is a species with respect to the next genus up and a genus with respect to the next species down. The qualities meant by the adjectives Corporeal, Animate, Sensible (that is, capable of feeling), and Rational are the successive differences that divide each genus into species. Notice that the negative parts of the genera, namely Incorporeal Substance, Inanimate Body, and so on, could also be subdivided. That has not been done, to avoid cluttering the figure.

Logical division is the name for the process by which we pick out the species a genus is made up of. We are said to divide the genus "book," for instance, when we think of it as made up of the groups folio, quarto, octavo, duodecimo books, and so on. The size of the books is in this case the ground, basis, or principle of division, usually called the Fundamentum Divisionis. For a quality or circumstance to serve as the basis of a division, it must be present in some members and absent in others, or it must vary among the different species the genus contains. A generic property, of course, is present in the whole of the genus, so it can't be used for dividing. A sound and useful division must obey three rules:

1. The constituent species must exclude each other.

2. The constituent species, added together, must be equal to the genus.

3. The division must be founded on one principle or basis.

It would obviously be absurd to divide books into folio, quarto, French, German, and dictionaries. These species overlap: there may be French or German dictionaries that happen to be quarto or folio, and so belong to three different species at once. A division like this is called a Cross Division, because it uses more than one principle of division, and so the species cross one another and cause confusion. If I divided rectilineal figures into triangles, parallelograms, rectangles, and polygons of more than four sides, I would commit every possible fault in a single division. The species parallelogram and rectangle don't exclude each other, since all rectangles must be parallelograms. The species added together don't make up the whole genus rectilineal figure, since irregular four-sided figures that are not parallelograms have been left out. And there are three principles of division at work: the number of sides, the directions of those sides, and the angles they contain. But when we subdivide, treating each species as a genus that can be split up further, we may, and in fact must, use a new principle of division each time. So I can divide rectilineal figures according to the three principles mentioned above:

Here the principles of division are the number of sides and, for four-sided figures, whether their sides are parallel. Triangles can't be divided in this second respect. We can make a further division of parallelograms, taking the equality of the sides and the size of the angles as the principles, like this:

From a logical point of view, the most perfect divisions come from dividing each genus again and again into two species by a difference. The Tree of Porphyry is an example of this. The process is called Dichotomy (from the Greek δίχα, dicha, in two, and τέμνω, temno, to cut). It is also called Exhaustive Division, because it always and necessarily obeys the second rule and provides a place for every possible thing. By a Law of Thought that we will consider in Lesson 14, everything must either have a given quality or not have it, so it must fall into one part or the other of the genus. This process of exhaustive division will turn out to be quite important in Lesson 23. In practice, though, it is by no means always necessary or convenient. For instance, it would make a needlessly long classification if we divided rectilineal figures like this:

We know beyond all doubt that every figure must have 3, 4, 5, 6, or more sides, and that no figure can belong to more than one of these groups. So it is much better to list the parts straight away: Triangle, Quadrilateral, Pentagon, Hexagon, and so on. Again, it would be very awkward to divide the counties of England into Middlesex and not-Middlesex, the second of these into Surrey and not-Surrey, and that again into Kent and not-Kent. Dichotomy is useless in cases like these, and even looks absurd, because a much shorter division lets us obey the rules of division with certainty. But in less certain branches of knowledge, our divisions can never be safe from a possible oversight unless they proceed by dichotomy. Suppose we divide the population of the world into three branches, Aryan (Indo-European), Semitic, and Turanian, as the scholars of the 1870s did. Some race might eventually be discovered that is distinct from all of these, and for which no place has been provided. But suppose we had proceeded like this:

Then the new race would obviously fall into the last group, the one that is neither Aryan, Semitic, nor Turanian. All the divisions of naturalists are exposed to this inconvenience. If we divide Vertebrate Animals into Mammalia, Birds, Reptiles, and Fish, a new form might be discovered at any time that belongs to none of these, and so upsets the division.

One more precaution is needed in Division: don't go straight from a high or wide genus to a low or narrow species. As the saying goes, divisio non faciat saltum, "the division should not make a leap." The species should always be those of the proximate genus, the genus immediately above them. It would obviously be awkward, for instance, to begin by dividing geometrical figures into those that have parallel sides and those that don't. Yet this principle of division is perfectly proper when it is applied to the proximate genus.

Logical division must not be confused with physical division, or Partition, in which an individual object, such as a tree, is thought of as made up of its separate parts: root, trunk, branches, leaves, and so on. There is even a third, distinct process, called Metaphysical Division, which consists in thinking of a thing as a collection of qualities and separating these in our minds, as when we distinguish the shape, color, taste, and smell of an orange.

Closely connected with the subject of this Lesson is the process of Logical Definition, by which we settle the common qualities, or marks, of the objects that belong to a given class. A definition must give the briefest possible statement of the qualities that are enough to distinguish the class from other classes and to fix its position in the general classification of our ideas. We can do this by treating the class as a species and giving the proximate genus and the difference. Here the word genus is used in its intensive meaning: it stands for the qualities that belong to everything in the genus and are enough to mark those things out. Since the difference then marks out the part of the genus we want, we get a perfect definition of the species. But we should be careful not to put any superfluous marks into a definition. If the extra marks are accidents and do not belong to the whole class, the definition is wrongly narrowed, as it would be if we defined Quadrilateral Figures as figures with four equal sides. If the extra marks belong to everything being defined, they are Properties, and they make no difference to the definition at all. Suppose I define parallelograms as "four-sided rectilineal figures, with the opposite sides equal and parallel, and the opposite angles equal." I have added two properties, the equality of the opposite sides and the equality of the opposite angles. These follow necessarily from the sides being parallel, and they only make the definition more complicated without making it any more precise.

Logic books usually give certain rules that sum up the precautions needed in definition.

1. A definition should state the essential attributes of the species defined. As far as the expression "essential attributes" can be given any exact meaning, it means, as explained above, the proximate genus and the difference.

2. A definition must not contain the name defined. The purpose of a definition is to make the species known, and as long as the species is not known, it can't serve to make itself known. When this rule is broken, there is said to be circulus in definiendo, or "a circle in defining," because the definition brings us right back round to the word we started from. This fault is usually committed by putting into the definition a word that is really a synonym of the name being defined. Examples would be defining "Plant" as "an organized being possessing vegetable life," or elements as simple substances, since vegetable is really equivalent to plant, and simple to elementary. If I defined metals as "substances possessing metallic luster," I would either be committing this fault, or else using the term metallic luster in a sense that would let in other substances too, and so breaking the next rule.

3. The definition must be exactly equivalent to the species defined. That is, it must be an expression whose denotation is neither narrower nor wider than the species, so that it takes in exactly the same objects. In short, the definition must denote the species, the whole species, and nothing but the species, and this can really be taken as a description of what a definition is.

4. A definition must not be expressed in obscure, figurative, or ambiguous language. In other words, every term used in the definition must be exactly known. Otherwise the purpose of the definition, which is to make us acquainted with the sufficient marks of the species, is obviously defeated. There is no worse logical fault than to define ignotum per ignotius, the unknown by the still more unknown. Aristotle's definition of the soul as "The Entelechy, or first form of an organized body which has potential life," certainly seems open to this objection.

5. Last, a definition must not be negative where it can be affirmative. This rule, though, often does not apply, and it is by no means always binding.

On the nature of Classification and the five Predicables, read Mr. J. S. Mill, System of Logic, Book I. Chap. VII. For the ancient Scholastic views on Definition, see Mansel's Artis Logicae Rudimenta (Aldrich), App. Note C.

Questions and Exercises

1. Define each of the five predicables.

2. In what sense can we say that the genus is part of the species, and in what sense that the species is part of the genus?

3. From the terms in question 6 of Lesson 5, pick out the ones that are genera, species, highest genera, or lowest species of other terms.

4. Explain the expressions sui generis, homogeneous, heterogeneous, summum genus, infima species, Tree of Porphyry.

5. Name a property and an accident of each of the following classes: Circle, Planet, Bird, Member of Parliament, Ruminant Animal.

6. What are the rules of correct logical division?

7. The first name in each of the following series of terms is a class. Divide and subdivide it so as to include all the smaller classes listed after it, in accordance with the laws of division.

	(1) People: Laity, Aliens, Naturalized Subjects, Peers, Natural-born Subjects, Clergy, Baronets, Commons.
	(2) Triangle: Equiangular, Isosceles, Right-angled, Scalene, Obtuse-angled.
	(3) Reasoning: Induction (Imperfect), Deduction, Mediate Inference, Induction, Hypothetical Syllogism, Disjunctive Syllogism.

8. Divide any of the following classes: Governments, Sciences, Logical terms, Propositions.

9. What does a logical definition consist of?

10. What are the rules of correct definition?

11. Which rules do the following definitions break?

(1) Life is the sum of the vital functions.

(2) Genus is the material part of the species.

(3) Illative conversion is that in which the truth of the converse can be inferred from that of the convertend.

(4) Mineral substances are those which have not been produced by the powers of vegetable or animal life.

(5) An equilateral triangle is a triangle whose sides and angles are respectively equal.

(6) An acute-angled triangle is one which has an acute angle.

Lesson 13: Pascal and Descartes on Method

It is doubtful whether anyone has ever had a sharper or more perfect mind than Blaise Pascal. He was born in 1623 at Clermont, in Auvergne, and showed signs of a remarkable character from his earliest years. At first his father tried to stop him studying geometry. But Pascal's genius for the subject, and his love of it, were so strong that by the age of twelve he had worked out many of the propositions of Euclid's first book by himself, with no help from any person or any book. It is hard to say what he deserves most admiration for: his mathematical discoveries, his invention of the first calculating machine, his wonderful Provincial Letters written against the Jesuits, or his profound Pensées, or Thoughts, a collection of his reflections on scientific and religious subjects.

Among these Thoughts is a remarkable fragment on logical method, and the substance of it also appears in the Port Royal Logic. It is the second article of the Pensées, and its title is Réflexions sur la géométrie en général (Reflections on Geometry in General). I know of no piece of writing that comes closer to perfect truth and perfect clearness of expression. So in this lesson I am going to give a free translation of the most important parts of this fragment, and add to it the rules of method from the Port Royal Logic and from Descartes's famous Essay on Method. Pascal's words are nearly as follows.

"The true method would give us demonstrations of the highest excellence, if only it were possible to use it fully. It consists in keeping two main rules. The first rule is never to use any term whose meaning we have not clearly explained. The second rule is never to put forward any proposition that we cannot demonstrate from truths already known. In a word: define all the terms, and prove all the propositions. But if I am to keep the rules of the very method I am explaining, I must first say what is to be understood by Definition.

"In geometry we recognize only the definitions that logicians call Nominal Definitions. These are definitions that give a name to things that have been clearly pointed out in terms that are perfectly understood. Those are the only definitions I am talking about."

What such definitions are worth, and what they are for, is to make what we say clearer and shorter. They let us "express in the one name we give what could otherwise be said only in several words. The condition is that the name we give must be stripped of any other meaning it might have, so that it carries only the meaning we intend it to stand for.

"For example, suppose we need to tell apart the numbers that can be divided into two equal parts from those that cannot. To avoid repeating this distinction over and over, we give it a name, like this: we call every number that can be divided into two equal parts an Even Number.

"This is a geometrical definition. We have first clearly pointed out a thing, namely any number that can be divided into two equal parts. Then we give it a name, stripped of every other meaning the name might have, so that it carries only the meaning we have pointed out.

"It follows that definitions are entirely free, and that they can never be contradicted. Nothing could be more permissible than giving any name we like to a thing we have clearly pointed out. We need only take care not to abuse this freedom to name things by giving the same name to two different things. Even that would be permissible, as long as we did not mix up the results and carry what is true of one over to the other. But if we do fall into this fault, we have a sure and unfailing remedy. It is to put the definition, in our minds, in place of the thing defined. We must keep the definition so constantly in mind that every time we speak of an even number, say, we understand exactly that it is a number divisible into two equal parts. The two must be so joined and inseparable in our thinking that whenever one is spoken of, the mind goes straight to the other.

"Geometers, and everyone who works methodically, give names to things only to make what they say shorter, not to reduce or alter their ideas of the things they are talking about. They expect the mind always to fill in the whole definition behind the short terms, which they use only to avoid the confusion that a mass of words would cause.

"Nothing stops the cunning fallacies of the sophists faster or more effectively than this method. We should always use it, and it is enough by itself to clear away every kind of difficulty and equivocation.

"With these things well understood, I return to my explanation of the true method, which, as I said, consists in defining everything and proving everything.

"This method would certainly be an excellent one, if it were not absolutely impossible. Plainly, the first terms we tried to define would need earlier terms to explain them. In the same way, the first propositions we tried to prove would depend on other propositions that came before them in our knowledge. So it is clear that we would never reach the first terms or the first propositions.

"So as we push our inquiries further and further back, we necessarily arrive at basic words we cannot define, and at principles so clear that we cannot find any clearer ones to prove them by. It follows that human beings are by nature, and unavoidably, unable to treat any science whatever by a perfect method. But it does not follow that we should give up every kind of method. The most perfect method open to us is neither to define everything and demonstrate everything, nor to define nothing and demonstrate nothing. It is to take the middle course: not to define things that are clear and understood by everyone, but to define all the rest; and not to prove truths that everyone knows, but to prove all the rest. People go wrong in this method both ways: those who try to define and prove everything, and those who fail to do it for things that are not self-evident."

This admirable passage makes one thing plain: as long as we use words, we can never avoid appealing in the end to things themselves. Every definition of a word needs one or more other words, and those words need defining too, and so on forever. Nor may we ever circle back to words we have already defined. If we define A by B, and B by C, and C by D, and then D by A, we commit what may be called a circulus in definiendo, a "circle in defining." This is a very serious fallacy, because it can lead us to think we know the nature of A, B, C and D when in fact we know nothing about them at all.

Pascal summed up his view of the geometrical method clearly in the following rules, which he put into the Port Royal Logic.

Footnote: Mr. Spencer Baynes's translation, p. 317.

1. To admit no terms in the least obscure or equivocal without defining them.

2. To employ in the definitions only terms perfectly known or already explained.

3. To demand as axioms only truths perfectly evident.

4. To prove all propositions which are at all obscure, by employing in their proof only the definitions which have preceded, or the axioms which have been accorded, or the propositions which have been already demonstrated, or the construction of the thing itself which is in dispute, when there may be any operation to perform.

5. Never to abuse the equivocation of terms by failing to substitute for them, mentally, the definitions which restrict and explain them.

The reader will easily see that these rules are much easier to lay down than to follow. Even geometers do not agree about which are the simplest axioms to assume, or which are the best definitions to make. There are many different opinions about the true definition of parallel lines, and about the simplest assumptions to make about their nature. If that is so in geometry, how much harder must it be to follow Pascal's rules with any confidence in the less certain branches of science? After geometry, mechanics is perhaps the most perfect science. Yet the best authorities have been far from agreeing on the exact definitions of such ideas as force, mass, moment, power and inertia, and very different opinions are still held about the simplest axioms from which the law of the composition of forces can be proved. All the same, if we keep steadily in mind, while studying each science, that we need to define every term as far as we can, and to prove every proposition that can be proved from a simpler one, we will do a great deal to clear away error and confusion.

I also want to give here the rules that the famous Descartes proposed for guiding the reason toward the truth. They are as follows:

1. Never to accept anything as true unless we clearly know it to be true. That is, we must carefully avoid haste and prejudice, and include nothing more in our judgments than what presents itself to the mind so clearly and distinctly that we have no room at all to doubt it.

2. To divide each difficulty we examine into as many parts as possible, or as many as are needed to solve it.

3. To direct our thoughts in an orderly way, starting with the simplest objects and the easiest to know, so as to climb step by step to knowledge of the most complex.

4. In every case, to make our lists so complete and our reviews so wide that we can be sure we have left nothing out.

Descartes first stated these rules in his admirable Discourse on Method, where he sets out his thoughts on the right way to conduct the reason and to search for truth in any of the sciences. This little book is easy to get hold of in the original French, and it has also been translated into English by Mr. Veitch. I strongly advise the reader to study it. It is impossible always to follow the rules of Descartes and Pascal, or even to know whether we are following them properly in every case. But it must still be valuable to know what we ought to aim at.

Footnote: Published at Edinburgh in 1850.

Read Locke's short Essay on the Conduct of the Understanding, which contains admirable remarks on how to acquire exact and logical habits of thought.

Questions and Exercises

(1) What is the use of nominal definitions?

(2) How must we use definitions in order to avoid confusion?

(3) How far can we be said to be free to use any name for any object?

(4) According to Pascal, what is the true method of avoiding error?

(5) How do we learn the meanings of words that cannot be defined?

(6) Give examples of words that can be clearly defined, and of others that cannot.

(7) State the five rules of method given in the Port Royal Logic.

(8) Explain Descartes's rules for reaching the truth.

Lesson 14: The Laws of Thought

The next lessons deal with the most common form of reasoning, known as the syllogism. Before the reader goes on to them, it is worth paying careful attention to the very simple laws of thought on which all reasoning must ultimately rest. These laws describe the very simplest truths: truths on which everyone must agree, and which at the same time apply to every notion we can conceive. We cannot think correctly, or avoid plain self-contradiction, unless we obey what are called the Three Primary Laws of Thought. They can be stated like this:

1. The Law of Identity. Whatever is, is.

2. The Law of Contradiction. Nothing can both be and not be.

3. The Law of Excluded Middle. Everything must either be or not be.

Stated this way, these laws may seem absurdly obvious, and Locke and others made fun of them for exactly that reason. But I have found that students are seldom able to see their full meaning and importance at first. In Lesson 23 we will see that until recent years logicians overlooked a very simple way of explaining all arguments, once these self-evident laws are granted. It is not too much to say that the whole of logic will be plain to anyone who constantly uses these laws as the key.

The first law may be taken as the best definition we can give of identity, or sameness. If anyone did not know what the word Identity meant, it would be enough to tell him that everything is Identical with itself.

The second law needs more thought. It means that nothing can have contradictory and inconsistent qualities at the same time and in the same place. A piece of paper may be blackened in one part while it is white in other parts. Or it may be white at one time and later become black. But we cannot conceive of it being both white and black in the same place at the same time. A door that has been open may be shut, but it cannot be shut and open at once. Water may feel warm to one hand and cold to the other, but it cannot be both warm and cold to the same hand. No quality can be both present and absent at the same time, and this seems to be the simplest and most general truth we can assert about all things. It is the very nature of existence that a thing cannot be other than it is. We can safely say that all fallacy and error come from reasoning, without realizing it, in a way that is inconsistent with this law. Any statement or inference that implies a combination of contradictory qualities must be taken as impossible and false, and breaking this law is the mark of its falseness. It is easy to show that if Iron is a metal, and every metal is an element, then Iron must be an element, or else it can be nothing at all, since it would combine qualities that are inconsistent (see Lesson 23).

The Law of Excluded Middle is much less self-evident than the other two, and the reader may not see at first that it is just as important and necessary as they are. The best way to explain it is this: it is impossible to name any thing, and any quality or circumstance, without granting that the quality or circumstance either belongs to the thing or does not belong to it. The name of the law expresses the fact that there is no third, or middle, course; the answer must be Yes or No. Let the thing be rock and the quality hard; then rock must be either hard or not-hard. Gold must be either white or not white; a line must be either straight or not straight; an action must be either virtuous or not virtuous. In fact, even when we know nothing about the terms we are using, we can still make assertions about them in agreement with this law. The reader may not know, and in fact chemists may not really know for certain, whether vanadium is a metal or not a metal. But anyone knows that it must be one or the other. Some readers may not know what a cycloid is, or what an isochronous curve is, but they must know that a cycloid either is an isochronous curve or is not an isochronous curve.

Still, the law of excluded middle is not so evident that no plausible objections can be raised against it. Someone might argue that rock is not always either hard or soft, because it may be halfway between, a little hard and a little soft at the same time. This objection points to a distinction of great logical importance, and neglecting it often leads to fallacy. The law of excluded middle said nothing about hard and soft; it spoke only of hard and not-hard. If the reader chooses to put soft in place of not-hard, he falls into a serious confusion between opposite terms and contradictory terms. A thing may quite possibly be neither hard nor soft, being halfway between. But in that case it cannot fairly be called hard, so the law still holds. In the same way, water must be either warm or not-warm, but it does not follow that it must be warm or cold. The alternative not-warm plainly includes all the cases where it is cold, and also the cases where it is at a middling temperature, so that we would call it neither warm nor cold. So we must carefully separate questions of degree, or quantity, from questions of simple logical fact. Where a thing or a quality can exist to a greater or lesser extent, there are many alternatives. Warm water, for instance, may have any temperature from perhaps 70° up to 120° (Fahrenheit). Exactly the same question comes up in geometrical reasoning. Euclid, in his Elements, often argues from the self-evident truth that any line must be either greater than, equal to, or less than any other line. While logic has only two alternatives to choose from, mathematics has three. So one line, compared with another, may be:

Another objection to the third law of thought is even more plausible. Take virtue as the thing and triangular as the quality. The Law of Excluded Middle lets us assert at once that virtue is either triangular or not-triangular. At first sight it might seem false and absurd to say that an immaterial notion like virtue is either triangular or not. Virtue has nothing in common with the material substances that occupy space, which are the only things the notion of shape belongs to. But the absurdity does not come from any falseness in the law. It comes from misreading the expression not-triangular. If, when we say that a thing is "not triangular," we are taken to imply that it has some shape, only not a triangular one, then of course the expression cannot be applied to virtue or to anything immaterial. In strict logic, however, no such implied meaning is allowed. Not-triangular includes both the things that have some shape other than triangular and the things that have no properties of shape at all; and it is in this second sense that it applies to an immaterial thing.

So these three laws are universally and necessarily true, whatever things they are applied to, and they become the foundation of reasoning. Every act of reasoning starts from certain judgments. An act of judgment consists in comparing two things or ideas and finding out whether they agree or differ, that is, whether they are identical in any of their qualities. The laws of thought tell us the very nature of this identity, which is what all thought is concerned with. But in the work of discourse, or reasoning, we need some additional laws, or axioms, or self-evident truths. They can be stated like this:

1. Two terms agreeing with one and the same third term agree with each other.

2. Two terms of which one agrees and the other does not agree with one and the same third term, do not agree with each other.

These self-evident truths are usually called the Canons, or Fundamental Principles, of Syllogism, and they are true whatever kind of agreement we are dealing with. The example we used in Lesson 1, where the terms "most useful metal" and "cheapest metal" agreed with the common third term "iron," was simply an instance of the first Canon, and there the agreement was complete identity. In the case of "the earth," "planets," and "bodies revolving in elliptical orbits," the agreement was less complete. The earth is only one of many planets, and the planets are only a small part of all the heavenly bodies that revolve in such orbits, which also include satellites, comets, meteors and double stars.

The second Canon applies to cases where there is disagreement, or difference, as in this example:

	Venus is a planet.
	Planets are not self-luminous.
	Therefore Venus is not self-luminous.

The first of these propositions states that a certain agreement exists between Venus and planet, just as in the earlier case of the earth. But the second proposition states a disagreement between planet and self-luminous bodies. From these we infer a disagreement between Venus and self-luminous body. But the reader should notice carefully that from two disagreements we can never infer anything. If the following were put forward as an argument, it would be plainly absurd:

	Sirius is not a planet.
	Planets are not self-luminous.
	Therefore Sirius is not self-luminous.

Both of the premises, or propositions given, are true, and yet the conclusion is false: all the fixed stars, Sirius among them, are self-luminous, which means they shine by their own light. In fact we may state a third Canon:

3. Two terms both disagreeing with one and the same third term may or may not agree with each other.

Self-evident rules of exactly the same kind as these three Canons are the basis of all mathematical reasoning, where they are usually called axioms. Euclid's first axiom is that "Things which are equal to the same thing are equal to one another;" and it holds true whether we apply it to the lengths of lines, the sizes of angles, areas, solids, numbers, degrees, or anything else that can be equal or unequal. So if the lines A and B are each equal to C, it is evident that each is equal to the other.

Euclid gives no axioms matching the second and third Canons, but they are really used in geometry. If A is equal to B, but D is not equal to B, it follows that A is not equal to D. In other words, things of which one is equal and the other unequal to the same third thing are unequal to each other. Lastly, A and E are two lines that are both unequal to D and unequal to each other, while A and B are two lines that are both unequal to D but equal to each other. So we can plainly see that "two things both unequal to the same thing may or may not be equal to each other."

From all this it will be clear that every piece of reasoning needs at least one agreement. If there are two agreements, we can reason to a third agreement. If there is one agreement and one difference, we can reason to a second difference. But if there are only two differences, we cannot reason to any conclusion at all. In the next lesson these self-evident principles will help explain some of the rules of the syllogism.

Logicians, however, have not kept to these Canons alone. They have often put the same truth into a different form, in an axiom called the Dictum de omni et nullo of Aristotle. This famous Latin phrase means "the statement about all and none," and the axiom, or rather pair of axioms, is usually given in these words:

Whatever is predicated of a term distributed whether affirmatively or negatively, may be predicated in like manner of everything contained under it.

Or more briefly:

What pertains to the higher class pertains also to the lower.

In plain, untechnical language this just means two things. First, whatever can be said of all the things of some sort or kind can be said of any one of them, or any part of them. Second, whatever can be denied of all the things in a class can be denied of any one of them, or any part of them. Whatever can be said of "all planets" can be said of Venus, the earth, Jupiter or any other planet. All the planets can be said to revolve in elliptical orbits, so this can be asserted of Venus, the earth, Jupiter or any other planet. In the same way, by the negative half of the Dictum, we can deny that the planets are self-luminous; and knowing that Jupiter is a planet, we can deny that Jupiter is self-luminous. A little thought will show that the affirmative Dictum is really the first Canon in a less complete and less general form, and that the negative Dictum is, in the same way, the second Canon. In fact these Dicta apply only to those cases of agreement between terms where one term is the name of a smaller class and the other is the name of a larger class that contains it. Logicians have mostly, and strangely, overlooked the important cases where one term agrees with another so far as to be identical with it. But this is not the place to discuss that subject at any length. It is treated in my little book called The Substitution of Similars.

Footnote: Macmillan and Co. 1869.

Some logicians have held that besides the three laws called the Primary Laws of Thought, there is a fourth, called "the Principle, or Law, of Sufficient Reason." Leibniz stated it in these words:

Nothing happens without a reason why it should be so rather than otherwise. For example, suppose there is a pair of scales, exactly alike in every respect on each side, with exactly equal weights in each pan. It must stay motionless and in balance, because there is no reason why one side should go down rather than the other. It is certainly a basic assumption of the science of mechanics that if a body is pushed by two perfectly equal forces in different directions, it will move evenly between them, because there is no reason why it should move more to one side than to the other. Mr. Mansel, Sir William Hamilton and others, however, think that this law has no place in logic, even if it can be held to be self-evident at all. The question seems open to doubt, and we need not discuss it here.

I have used the word axiom so freely in this lesson that I should make its meaning as clear as I can. Philosophers do not entirely agree about where it comes from or exactly what it means, but it certainly comes from the Greek verb ἀξιόω (axioō), which is translated "to think worthy." It generally means a self-evident truth so simple that it must be assumed to be true. Since it cannot be proved from any simpler proposition, it must itself be taken as the basis of reasoning. In mathematics it is clearly used in this sense.

See Hamilton's Lectures on Logic, Lectures 5 and 6.

Questions and Exercises

1. State the three Fundamental Laws of Thought, and apply them to the following notions:

(1) Matter, organic, inorganic.

(2) Undulations, polarized, non-polarized.

(3) Figure, rectilinear, curvilinear.

2. Is it wrong to assert that an animal cannot be both vertebrate and invertebrate, given that some animals are vertebrate and some are not?

3. From the following terms, pick out those that are negatives of the others, and those that are opposites: light, plenum, gain, heat, decrease, loss, darkness, cold, increase, vacuum.

4. How does Aristotle's dictum apply to the following arguments?

(1) Silver is a good conductor of electricity; for such are all the metals.

(2) Comets cannot be without weight; for they are composed of matter, which is not without weight.

Lesson 15: The Rules of the Syllogism

Syllogism is the common name for Mediate Inference: inference through a medium, or middle term. It is to be distinguished from Immediate Inference, which is inference carried out without any third, or middle, term.

We use a middle term, or medium, whenever we can't compare two things with each other directly but can compare each of them with some third thing. We can't compare the sizes of two halls by putting one inside the other. But we can measure each of them with a foot rule or some other suitable measure. That measure is common to both, and it lets us find out their relative size as accurately as we need. Or suppose we have two lots of cotton cloth and want to compare them. We don't have to carry the whole of one lot over to the other. We cut off a sample that represents exactly the quality of the first lot, and the two lots agree or differ according to whether the sample agrees with the second lot or not.

A middle term in a syllogism works very much like the medium in these cases, though not exactly the same way. Suppose, for example, that we want to find out whether or not "whales are viviparous" (that is, give birth to live young), and we have no chance to observe the fact directly. We could still show that it is so if we knew that "whales are mammalian animals" and that "all mammalian animals are viviparous." It would follow that "whales are viviparous." And as far as the inference is concerned, it makes no difference what meaning we give to the words viviparous and mammalian. In this case "mammalian animal" is the middle term.

The name Syllogism means the joining together of two propositions in thought. It comes from the Greek words σύν (syn), together, and λόγος (logos), thought. So it is exactly equivalent to the word Computation, which means thinking together (Latin con, together, and puto, to think), or reckoning. In a syllogism we unite in thought two premises, or propositions put forward, in such a way that we can draw from them, or infer, a third proposition, called the conclusion, by means of the middle term they contain. So we may define Syllogism as the act of thought by which we go from two given propositions to a third proposition, whose truth necessarily follows from the truth of the two given ones. When the argument is fully expressed in words, it is usual to call it, concretely, a syllogism.

The special rules of the syllogism are founded on the Laws of Thought and the Canons we looked at in the last lesson. They tell us exactly when one proposition can be inferred from two others. There are eight of them, as follows:

1. Every syllogism has three and only three terms.

These terms are called the major term, the minor term, and the middle term.

2. Every syllogism contains three, and only three, propositions.

These propositions are called the major premise, the minor premise, and the conclusion.

3. The middle term must be distributed at least once, and must not be ambiguous.

4. No term may be distributed in the conclusion that was not distributed in one of the premises.

5. From negative premises nothing can be inferred.

6. If one premise is negative, the conclusion must be negative; and the other way around, to prove a negative conclusion one of the premises must be negative.

From these rules we can deduce two subordinate rules, which it will be convenient to state right away all the same.

7. From two particular premises no conclusion can be drawn.

8. If one premise is particular, the conclusion must be particular.

All these rules are so extremely important that the student should not only understand perfectly what they mean and why they are true, but also learn them by heart. In the rest of this lesson we will look at what they mean and what force they have.

Since the syllogism consists in comparing two terms by means of a middle term, there obviously can't be fewer than three terms. Nor can there be more. Suppose there were four terms, say A, B, C, D, and we compared A with B and C with D. Then either we would have no common medium at all between A and D, or we would need a second syllogism: first comparing A and C with B, and then A and D with C.

You can always recognize the middle term by the fact that it does not appear in the conclusion. The major term is always the predicate of the conclusion, and the minor term is always the subject. They get these names because in the universal affirmative proposition (A) the predicate is necessarily a wider, or greater, or major term than the subject. In "all humans are mortals," for example, the predicate takes in all the other animals as well as humans, so it is obviously a major, or wider, term than humans.

Next, the syllogism necessarily consists of three parts. There is the major premise, in which the major and middle terms are compared. There is the minor premise, which in the same way compares the minor and middle terms. And there is the conclusion, which contains only the major and minor terms. In a strictly correct syllogism the major premise always comes before the minor premise, but in ordinary writing and speech this rule is seldom kept. Whichever premise contains the major term is still the major premise, wherever it stands.

The third rule is a very important one, because many fallacies come from ignoring it. When we say the middle term must be distributed at least once, we mean (see Lesson 9) that the whole of it must be referred to universally in one premise, if not in both. Take the two propositions

	All Frenchmen are Europeans,
	All Russians are Europeans.

They do not distribute the middle term at all, because both are affirmative propositions, and affirmative propositions have undistributed predicates (see Lesson 9). Plainly, Frenchmen are one part of the Europeans and Russians are another part, as Euler's method shows in Fig. 6, so there is no real middle term. These propositions would allow Russians to be Frenchmen just as well as not to be. Whether the two inner circles overlap or not, both are still inside the larger circle of Europeans. Or take the two propositions

	All Frenchmen are Europeans,
	All Parisians are Europeans.

They do not let us infer that all Parisians are Frenchmen. Of course we know that all Parisians are included among Frenchmen, but the premises would let them be placed anywhere inside the circle of Europeans. This example shows that the premises and the conclusion of an apparent argument may all be true, and the argument may still be fallacious.

The part of the third rule about an ambiguous middle term hardly needs explaining. We said in Lesson 4 that an ambiguous term is one with two different meanings, implying different connotations. It is really the same as two different terms that happen to be spelled the same way, so that they are easily mistaken for each other. Suppose we argued that "all metals are elements, and brass is metal, therefore it is an element." We would be committing a fallacy, because we would be using the middle term metal in two different senses. In one sense it means the pure, simple substances that chemists call metals. In the other it means a mixture of metals that is commonly called metal in the trades, but that chemists call an alloy. In many of the examples found in logic books, the ambiguity of the middle term is extremely obvious. But the reader should always be ready to meet cases where the ambiguity is extremely subtle and hard to see. For instance, someone might argue that "what is right should be enforced by law, and charity is right, and should therefore be enforced by the law." Here it is clear that right is used in one place for what the conscience approves, and in the other for what public opinion holds to be necessary for the good of society.

The fourth rule forbids us to distribute a term in the conclusion unless it was distributed in the premises. The only purpose of the syllogism is to prove the conclusion from the premises. So obviously we must not make a statement about anything unless that thing was mentioned in the premises in a way that justifies the statement. Suppose we argued that "because many nations are capable of self-government, and nations capable of self-government should not receive laws from a despotic government, therefore no nation should receive laws from a despotic government." We would plainly be going beyond what our premises contain. The minor term, many nations, was particular in the minor premise, and must not be made universal in the conclusion. The premises justify a statement only about the many nations that are capable of self-government. So this argument would be fallacious, and its technical name would be an illicit process of the minor term, meaning that we have treated the minor term improperly.

A breach of the fourth rule like this one is extremely easy to detect, and so it is very seldom committed.

But an illicit process, or improper treatment, of the major term is more common, because it is not so obviously false. Suppose we argued that "because all Anglo-Saxons love liberty, and Frenchmen are not Anglo-Saxons, therefore they do not love liberty." The fallacy would be fairly plain, but without knowing some logic it would not be easy to explain clearly what is wrong. The major term, loving liberty, is undistributed in the major premise, so we must take the Anglo-Saxons to be only a part of those who love liberty. So shutting Frenchmen out of the class of Anglo-Saxons does not necessarily shut them out of the class of those who love liberty (see Fig. 8). The conclusion of the false argument is negative, so it distributes its predicate, the major term. Since that term is undistributed in the major premise, we have an Illicit Major, as we may call this fallacy for short. Here is a less obvious example of the same fallacy: "Few students are capable of excelling in many branches of knowledge, and those who can excel in this way are deserving of high praise"; hence "few students are deserving of high praise." The little word "few" here has the double meaning explained earlier (see Lesson 8): it means "a few are, and so on, and the rest are not." So the conclusion is really a negative proposition, and it distributes the major term "deserving of high praise." But this major term is clearly undistributed in the major premise. That premise only says that those who can excel in many branches of knowledge are deserving of praise. It says nothing, and implies nothing, about the other students.

The fifth rule is clearly founded on the principle we noticed in the last lesson: inference can only go forward where there is agreement, and two differences, or disagreements, allow no reasoning at all. As the third Canon states, two terms may both differ from a common term and still may or may not differ from each other. Suppose we argued that Americans are not Europeans and Virginians are not Europeans. Both terms disagree with the middle term Europeans, and yet they agree with each other. In other cases the two negative premises may be plainly true while it stays quite uncertain whether the major and minor terms agree or not. It is true, for instance, that "colonists are not Europeans, and Americans are not Europeans." But this gives us no right to infer either that colonists are Americans or that they are not. Fig. 9 represents the two negative premises by keeping the circles of Colonists and Americans outside the circle of Europeans. But they can be kept outside it whether Colonists and Americans coincide partly, or wholly, or not at all. A breach of this rule of the syllogism may conveniently be called the fallacy of negative premises. We must not suppose, though, that just having a negative word (not or no) in a proposition makes it negative in the sense this rule means. Take the argument

	"What is not compound is an element.
	Gold is not compound;
	Therefore Gold is an element."

It has negatives in both premises, but it is valid all the same, because in both cases the negative belongs to the middle term, which is really the negative term not-compound.

The truth of the sixth rule depends on the truth of the axiom that if two terms agree with a common third term, they agree with each other. Remember that a negative proposition asserts disagreement. Then it is clear that a negative conclusion could not be drawn from premises that are really affirmative. The matching negative axiom prevents us from drawing an affirmative conclusion if either premise is really negative. Only practice, however, will let the student apply this rule and the ones before it with confidence, since a fallacy can be hidden and disguised by all sorts of ways of putting things. The questions at the end of Lessons 18 and 21 give many examples on which the student can become skilled at analyzing arguments.

The remaining rules of the syllogism, the 7th and the 8th, are by no means self-evident. They are in fact corollaries of the first six rules, that is, consequences that follow from them. So we will have to show in Lesson 17 that they really do follow. We may call a breach of the 7th rule a fallacy of particular premises, and a breach of the 8th rule the fallacy of a universal conclusion from a particular premise. But these fallacies can really be reduced to those of Illicit Process or Undistributed Middle.

For many details about the Aristotelian and Scholastic views of the Syllogism, and of Formal Logic in general, see the full critical notes in Mansel's edition of Aldrich's Artis Logicæ Rudimenta. 2nd Ed. Oxford. 1852.

Questions and Exercises

1. Distinguish mediate and immediate inference.

2. Define syllogism, and say what word it is synonymous with.

3. What are the six principal and two subordinate rules of the syllogism?

4. In the following syllogisms, point out in turn the conclusion, the middle term, the major term, the minor term, the major premise and the minor premise, keeping to exactly this order.

(1) All human beings are fallible; All kings are human beings; Therefore all kings are fallible.

(2) Platinum is a metal; All metals combine with oxygen; Therefore Platinum combines with oxygen.

(3) The Khoikhoi are capable of education; for the Khoikhoi are human beings, and all human beings are capable of education.

5. Explain carefully what is meant by non-distribution of the middle term.

Lesson 16: The Moods and Figures of the Syllogism

We now have in hand all the principles of reasoning, and the rules built on them, that let us tell a true syllogism from one that only looks like one. Our job in this lesson is to find out all the different shapes in which a mediate inference, or syllogism, can turn up. We know that every syllogistic argument must contain three propositions, and three distinct terms, each appearing twice in those propositions. As far as we know so far, each proposition of the syllogism can be either affirmative or negative, and either universal or particular. So it is not hard to work out the largest possible number of ways a syllogism could conceivably be put together. In short, any one of the four propositions A, E, I or O can be taken as the major premise; it can be joined to any one of the same four as the minor premise; and any one of the four again can be added as the conclusion. That gives us a series of combinations, or ways of joining the letters A, E, I, O. Here are a few of them written out:

Clearly there will be 4x4x4, or 64, such combinations in all, and only 23 of them are given above. The student can easily write out the rest by carrying on the same orderly changes of the letters. Starting with AAA, we change the right-hand letter to E, then I, then O. Then we do the same starting from AEA instead. Once the middle letter has been carried through all its changes, we start changing the left-hand letter. With each change of the left-hand letter we have to repeat all sixteen changes of the other two, so there are plainly 64 different conceivable ways of arranging propositions into syllogisms.

We call each of these triplets of propositions a mood, or form, of the syllogism (from the Latin modus, shape). Now we have to work out how many of these forms can really be used in valid arguments, as opposed to those that break one or more of the rules of the syllogism. The mood AEA, for example, would break the 6th rule, that if one premise is negative the conclusion must be negative too. AIE breaks the other half of the same rule, that a negative conclusion can only be proved by means of a negative premise. EEA, EEE and so on break the 5th rule, which forbids us to reason at all from two negative premises. Examples of any of these moods are easy to invent, and their falsity is very obvious. For AEA we might take:

	All Austrians are Europeans,
	No Australians are Europeans;
	Therefore, all Australians are Austrians.

Many of the 64 conceivable moods are ruled out by the 7th and 8th rules of the syllogism. AIA and EIE break the rule that if one premise is particular, the conclusion must be particular too. IIA, IOO, OIO and many others break the rule against two particular premises. Some combinations break more than one rule. OOO has both negative premises and particular premises, and OOA breaks the 6th rule as well. It is an excellent exercise in using the rules of the syllogism to write out all 64 combinations and then strike out every one that breaks a rule. If you go about it systematically, the job will not be as long or as tedious as it might seem. You will find that only twelve moods escape being ruled out, and so far these can be considered good forms of reasoning. They are:

Of these, though, IEO will soon have to be thrown out, because we will find that it really breaks the 4th rule and involves an Illicit process of the major term. So there are only eleven moods of the syllogism that are really valid, and we can now account for all sixty-four moods:

We have by no means used up all the possible varieties of the syllogism yet. So far we have only settled the character of the propositions: affirmative or negative, general or particular. We have not settled the ways the terms can be arranged inside them. The major term must be the predicate of the conclusion, but it can be either the subject or the predicate of the major premise. In the same way, the minor term, the subject of the conclusion, can be either the subject or the predicate of the minor premise. This gives four different ways in which the terms can be arranged, and these are called Figures. The four figures of the syllogism are shown in the following scheme, taking

	X to denote the major term
	Y to denote the middle term
	Z to denote the minor term

These figures must be carefully learned by heart, and the best way to do it is to notice where the middle term stands. In the 1st Figure it stands first, as the subject of the major premise. In the 2nd Figure it stands second, as the predicate of both premises. In the 3rd Figure it stands first again, as the subject of both premises. In the 4th Figure it takes an in-between position. In the conclusion, of course, the major and minor terms always have the same fixed places, so once the middle term is correctly placed in any figure, the rest of the syllogism is easy to fill in.

The reader will hardly be pleased to hear that each of the eleven valid moods has to be examined in each of the four figures separately. That leaves 44 possible cases, and the valid syllogisms have to be picked out from among them. The mood AEE in the first figure, for example, would look like this:

	All Y's are X's,
	No Z's are Y's;
	Therefore no Z's are X's.

This would break the 4th rule and be an Illicit Major, because X is distributed in the conclusion, which is a negative proposition, but not in the major premise. In the second figure the same mood would be valid:

	All X's are Y's,
	No Z's are Y's;
	Therefore no Z's are X's.

In the third figure it becomes

	All Y's are X's,
	No Y's are Z's,
	No Z's are X's,

and once again it breaks the 4th rule with respect to the major term. Finally, in the 4th figure it is valid, as the reader can easily check for himself.

When all the valid moods have been picked out of the 44 possible ones, there turn out to be 24 altogether. Here they are:

	Valid Moods of the Syllogism.

Five of these moods are set apart in brackets because, although they are valid, they are of little or no use. They are said to have a weakened conclusion, because the conclusion is particular when a general one could have been drawn. AAI in the first figure, for example, is represented by this:

	All material substances gravitate,
	All metals are material substances;
	Therefore some metals gravitate.

Obviously the conclusion states only part of the truth: in reality all metals gravitate. It is not actually a wrong conclusion. As we must carefully remember from Lesson 9, affirming a subaltern, or particular, proposition does not deny the general proposition that goes with it. It is perfectly true that some metals gravitate, and it must be true, because all of them do. But when we can just as easily prove that all of them gravitate, that is the conclusion we ought to draw.

If we follow most logicians and set aside the five syllogisms with weakened conclusions, nineteen remain that are both valid and useful. In the next lesson we will be given some ancient memory-verses, and those verses are about the only means by which most people could keep these 19 combinations in their heads. In the meantime, the reader can see from the table of valid moods above that the following remarks about the particular character of each figure of the syllogism are true.

The first figure is the only one that proves the proposition A, that is, the only one with A as a conclusion. It is also the only figure that can prove every one of the four propositions A, E, I and O. As for its premises, it is especially important to notice that the major premise is always universal (A or E) and the minor premise is always affirmative (A or I). We will look at this peculiarity further in the next lesson.

The second figure proves only negative conclusions (E or O), and the reason is easy to see. In this figure the middle term is the predicate of both premises. So if both premises were affirmative, the middle term would be undistributed in both of them, and we would commit the fallacy of undistributed middle that we saw illustrated in Lesson 15 (the Frenchmen and the Russians). It follows that one premise must be negative, and of course only one. So of the major and minor terms, one must be wholly included in the middle term or wholly excluded from it, and the other must at the same time be excluded from it or included in it, at least partly. To illustrate this, let X, Y and Z stand, as before, for the major, middle and minor terms of a syllogism. The four moods of this figure are then

The nature of the moods of the second figure is shown clearly in the following figures:

Notice also that in the second figure the minor premise can be any of the four, A, E, I or O.

The third figure proves only particulars (I or O), and its minor premise is always affirmative (A or I). It also has the largest number of moods, because none of its conclusions is a weakened one.

The fourth figure is usually considered unnatural and fairly useless, because the same arguments can be set out more clearly in the form of the first figure, which it resembles in some ways. It proves every proposition except A, that is, E, I and O, and its first mood, AAI, is really a weakened form of AAA in the first figure. Many logicians, including Sir William Hamilton in recent times, have rejected this figure altogether.

Clearly the different figures of the syllogism have different characters, and logicians have thought that each figure is best suited to certain special purposes. A German logician, Lambert, stated these purposes briefly as follows: "The first figure is suited to discovering or proving the properties of a thing; the second to discovering or proving the differences between things; the third to discovering or proving instances and exceptions; the fourth to discovering, or ruling out, the different species of a genus."

We may add that the moods Cesare and Camestres are often used to disprove a statement, because they give a universal negative conclusion, based on shutting one class out of another. Suppose someone still claimed that light consists of material particles. We could answer him with the following syllogism:

	"Material particles pass on impetus to whatever they strike.
	Light does not pass on impetus to whatever it strikes;
	Therefore light is not material particles."

The moods Baroko and Festino are used less, but they allow a particular conclusion to be established.

When we want to raise objections or exceptions to a general statement, though, which is in fact the natural way of answering one, we use the third figure. The statement that "all metals are solids" would be disproved at once by the exception mercury, like this:

	Mercury is not solid,
	Mercury is a metal;
	Therefore some metal is not solid.

Suppose someone claimed that what is incomprehensible cannot exist. We answer at once with this argument: infinity is incomprehensible; but infinity certainly exists, because without it we cannot explain the nature of a curved line, or of a quantity that varies continuously; therefore something that is incomprehensible exists. In this case even one exception is enough to overturn the claim completely, since the claim really means that because a thing is incomprehensible it cannot exist. And if one incomprehensible thing does exist, others may too, and the statement loses all its authority.

Under the Aristotelian system, the third figure must also be used whenever the middle term is a singular term, because in Aristotle's view of the subject a singular term could not stand as the predicate of a proposition.

Questions and Exercises

1. Name the rules of the syllogism broken by each of the following moods, paying no attention to figure: AIA, EEI, IEA, IOI, IIA, AEI.

2. Write out all 64 moods of the syllogism and strike out the 53 invalid ones.

3. Show in which figures the following premises give a valid conclusion: AA, AI, EA, OA.

4. In which figures are IEO and EIO valid?

5. To which moods do the following valid syllogisms belong? Arrange them in correct logical order.

(1) Some Y's are Z's. No X's are Y's. Some Z's are not X's.

(2) All Z's are Y's. No Y's are X's. No Z's are X's.

(3) No fish suckles its young; The whale suckles its young; Therefore the whale is no fish.

6. Draw conclusions from the following premises, and say which mood each syllogism belongs to.

(1) Some amphibious animals are mammalian. All mammalian animals are vertebrate.

(2) All planets are heavenly bodies. No planets are self-luminous.

(3) Mammalian animals are quadrupeds. No birds are quadrupeds.

(4) Ruminant animals are not predacious. The lion is predacious.

7. Invent examples to show that false premises can give true conclusions.

8. Supply premises for the following conclusions:

(1) Some logicians are not good reasoners.

(2) The rings of Saturn are material bodies.

(3) Party government exists in every democracy.

(4) All fixed stars obey the law of gravitation.

Lesson 17: Reduction of the Imperfect Figures of the Syllogism

To make the nineteen valid and useful moods of the syllogism easier to remember, logicians invented, at least six centuries ago, a very curious system of artificial words, put together into verses that are easy to learn by heart. However clever the device is, it is barbarous and entirely unscientific. Still, a student of logic is expected to know how it is built and how to use it, so the verses are given and explained below:

	Barbara, Celarent, Darii, Ferioque, prioris;
	Cesare, Camestres, Festino, Baroko, secundæ;
	Tertia, Darapti, Disamis, Datisi, Felapton,
	Bokardo, Ferison, habet; Quarta insuper addit
	Bramantip, Camenes, Dimaris, Fesapo, Fresison.

The words that are not mood names are real Latin words. They tell us that the four moods whose artificial names are Barbara, Celarent, Darii and Ferio belong to the first figure ("prioris," of the first); that the next four belong to the second ("secundæ"); that the third ("Tertia") has six more; and that the fourth figure ("Quarta") adds five moods besides. Each artificial name contains three vowels, and these give the propositions that make up a valid mood. So CElArEnt stands for the mood of the first figure that has E for its major premise, A for its minor premise, and E for its conclusion. Taken all together, the artificial words contain exactly the combinations of vowels shown in the table of valid moods in Lesson 16, except those in brackets.

These verses also contain directions for proving each mood of the second, third and fourth figures by reducing it to a matching mood of the first figure. Aristotle regarded the first figure as an especially clear and convincing form of argument, because the Dictum de omni et nullo applies to it directly, and so he called it the Perfect Figure. He never recognized the fourth figure at all. It is often called the Galenian figure, because the famous Greek physician Galen is supposed to have discovered it. Aristotle knew the second and third figures as the Imperfect Figures. They had to be reduced to the first figure by certain conversions of the premises and by changing the order of the premises, and the artificial words contain the directions for doing this. The directions are as follows:

The letter s means that the proposition marked by the vowel just before it is to be converted simply.

The letter p means that the proposition is to be converted per accidens, or by limitation.

The letter m means that the premises of the syllogism are to be transposed: the old major premise becomes the minor premise of a new syllogism, and the old minor becomes the new major. The m comes from the Latin mutare, to change.

B, C, D, F, the consonants the names begin with, tell us which mood of the first figure the reduction produces. So Cesare, Camestres and Camenes reduce to Celarent; Darapti and the others beginning with D reduce to Darii; Fresison reduces to Ferio; and so on.

The letter k means that the mood must be reduced, or proved, by a separate process called Indirect reduction, or reductio ad impossibile, which we will come to shortly.

Now let us take a syllogism, say one in Camestres, and follow the directions for reducing it. Let the example be:

	All stars are self-luminous (1)
	No planets are self-luminous (2)
	Therefore no planets are stars (3)

The first s in Camestres tells us to convert the minor premise simply. The m tells us to change the order of the premises, and the final s tells us to convert the conclusion simply. When we have made all these changes, we get:

	No self-luminous bodies are planets, converse of (2)
	All stars are self-luminous (1)
	Therefore no stars are planets, converse of (3)

This, as we can check, is a syllogism in Celarent, which we could have known from the initial C of Camestres.

For another example, let us take Fesapo:

	No fixed stars are planets,
	All planets are round bodies;
	Therefore some round bodies are not fixed stars.

The directions in the name tell us to convert the major premise simply, and the minor premise by limitation. We then have this syllogism in Ferio:

	No planets are fixed stars,
	Some round bodies are planets;
	Therefore some round bodies are not fixed stars.

The reader will find it easy to apply the same process of conversion or transposition to the other moods, following the directions in their names. The only moods we need to look at specially are Bramantip, Baroko and Bokardo. As an example of Bramantip we may take:

	All metals are material substances.
	All material substances are gravitating bodies;
	Therefore some gravitating bodies are metals.

The name contains the letter m, which tells us to transpose the premises, and the letter p, which means conversion by limitation. Making these changes, we have:

	All material substances are gravitating bodies,
	All metals are material substances;
	Therefore some metals are gravitating bodies.

This is not a syllogism in Barbara, as we might have expected. It is the weakened mood AAI of the first figure. Plainly these premises yield the conclusion "all metals are gravitating bodies," so in this mood we must take the letter p to mean that the conclusion is weaker than it could be. The truth is that the fourth figure is so imperfect and unnatural in form that Aristotle, the founder of logical science, never admitted that it existed at all. It contains nothing but badly arranged syllogisms, which would have been better stated in the first figure. It is a pity that such a needless addition was made to the forms of the syllogism, which are complicated enough already.

The two odd moods called Baroko and Bokardo cause a good deal of trouble, because they cannot be reduced directly to the first figure. To show how to handle them, we will let X, Y and Z stand for the major, middle and minor terms of the syllogism. Baroko can then be stated like this:

	All X's are Y's,
	Some Z's are not Y's;
	Therefore some Z's are not X's.

Now if we convert the major premise by contraposition (see Lesson 10), we get "all not-Y's are not X's." Making this the major premise of the syllogism, we have:

	All not-Y's are not X's,
	Some Z's are not-Y's;
	Therefore some Z's are not X's.

Both of these premises look negative, but this is really a valid syllogism in Ferio. Two of the negative particles only affect the middle term (see Lesson 15), and so we have carried out the reduction of the syllogism.

Stated in the same way, Bokardo is:

	Some Y's are not X's,
	All Y's are Z's;
	Therefore some Z's are not X's.

To reduce this, convert the major premise by negation, and then transpose the premises. We get:

	All Y's are Z's,
	Some not-X's are Y's;
	Therefore some not-X's are Z's.

This conclusion is the converse by negation of the original conclusion. So the truth of the original conclusion is proved by reducing the argument to a syllogism in Darii.

Both of these moods, Baroko and Bokardo, can also be proved by a special process of Indirect reduction, very much like the indirect proofs Euclid often uses in geometry. The process works like this. We suppose that the conclusion of the syllogism is false, and that its contradictory is therefore true. We can then easily build a new syllogism that leads to a conclusion contradicting one of the original premises. Now in logic it is absurd to call the truth of our own premises into question. The whole purpose of argument, or syllogism, is to draw a conclusion that will be true when the premises are true. The syllogism lets us restate, in a new form, the information already contained in the premises, just as a machine can hand back to us in a new form the material we put into it. The machine, or rather its maker, is not responsible for the quality of the materials fed into it. In the same way, the logician is not in the least responsible for the truth of his premises, only for handling them correctly. If he handles them at all, he must treat them as true. So a conclusion that requires one of our premises to be false is completely absurd.

To apply this method, we may take Baroko as before:

	All X's are Y's (1)
	Some Z's are not Y's (2)
	Therefore some Z's are not X's (3)

If this conclusion is not true, then its contradictory, "all Z's are X's," must necessarily be taken as true (see Lesson 9). Making this the minor premise of a new syllogism, with the original major premise, we have:

	All X's are Y's (1)
	All Z's are X's, contradictory of (3)
	Hence all Z's are Y's.

Now this conclusion, in A, is the contradictory of our old minor premise, in O. So we must either admit that one of our own premises is false or grant that our original conclusion is true. Of course the second is the one we choose.

We treat Bokardo in a very similar way:

	Some Y's are not X's (1)
	All Y's are Z's (2)
	Therefore some Z's are not X's (3)

If this conclusion is not true, then "all Z's are X's" must be true. Now we can make the syllogism:

	All Z's are X's, contradictory of (3)
	All Y's are Z's (2)
	Hence all Y's are X's.

This conclusion is the contradictory of (1), the original major premise. Since that cannot be allowed, we must either suppose that (2), the original minor premise, is false, which is just as impossible, or grant that our original conclusion is true.

Notice that in both of these cases of Indirect Reduction, or Indirect Proof, we use a syllogism in Barbara, and that is what the initial letters of Baroko and Bokardo indicate. The same process of indirect proof can be applied to any of the other moods, but it is not usual to do so, because the simpler process of direct reduction, often called ostensive reduction, is enough.

You will remember that when we looked at the rules of the syllogism toward the end of Lesson 15, there were two supplementary rules, the 7th and 8th, about particular premises. These were not at all self-evident, and they need to be proved from the six more fundamental rules. We have now come far enough to look at this proof with profit. The 7th rule forbids us to draw any conclusion from two particular premises. Two such premises must be II, IO, OI or OO. Of these, II contains no distributed term at all, so it must break the 3rd rule, which requires the middle term to be distributed. The premises OO plainly break the 5th rule, against negative premises. With the pair IO, the conclusion must be negative by the 6th rule, because one premise is negative. The major term will therefore be distributed in the conclusion. But the major premise is a particular affirmative, so the major term cannot be distributed there, and drawing the conclusion would commit the fallacy of illicit process of the major, against rule 4. Lastly, the premises OI contain only one distributed term, the predicate of the major premise. But since the conclusion must be negative by the 6th rule, the major term must be distributed. So we would need two distributed terms in the premises, one for the middle term and the other for the major term. The premises contain only one, so if we try to draw any conclusion at all, we must commit either the fallacy of undistributed middle or that of illicit process of the major term. We see, then, that in no possible case can a pair of particular premises give a valid conclusion.

The 8th rule of the syllogism tells us that if one premise of a syllogism is particular, the conclusion must also be particular. The only way to show this is true is to go through all the possible cases and observe that the six principal rules of the syllogism always require the conclusion to be particular. Suppose, for instance, the premises are A and I. Then they contain only one distributed term, the subject of A, and rule 3 requires it for the middle term. So the minor term cannot be distributed in the conclusion without breaking rule 4, and the conclusion must be the proposition I. The premises AO would contain two distributed terms, the subject of A and the predicate of O. But if we drew the conclusion E from them, both the major and the minor terms would have to be distributed, and the middle term would be left undistributed, against rule 3. The reader can easily prove the other cases, such as EI, by counting the distributed terms in the same way. It will always turn out that the premises do not distribute enough terms to allow a universal conclusion.

Questions and Exercises

1. State and explain the mnemonic lines Barbara, Celarent, and so on.

2. Construct syllogisms in each of the following moods, taking X, Y and Z for the major, middle and minor terms respectively, and show how to reduce them to the first figure:

Cesare, Festino, Darapti, Datisi, Ferison, Camenes, Fesapo.

3. What is the use of Reduction?

4. Prove that the following premises cannot give a universal conclusion: EI, IA, OA, IE.

5. Prove that the third figure must have an affirmative minor premise and a particular conclusion.

6. Reduce the moods Cesare and Camenes by the Indirect method, or Reductio ad Impossibile.

Lesson 18: Irregular and Compound Syllogisms

It may seem surprising, but the arguments we meet in books or in conversation are hardly ever set out in the form of regular syllogisms. Even when a complete syllogism does turn up now and then, it is usually there as a mere show of logical precision. In earlier centuries, it is true, every student at the universities had to take part in public disputations. In these, one side put forward elaborate syllogistic arguments and the other side refuted them with precise syllogisms of its own. The practice was given up at the University of Oxford not very long ago, and it is said to survive still at some universities on the Continent. But outside disputations of that kind, we have to admit, perfectly formal syllogisms are seldom used.

The truth is, though, that syllogistic arguments are not missing at all. Wherever one of the conjunctions therefore, because, for, since, hence, inasmuch as, consequently appears, we can be sure that an inference is being drawn, and very probably it is being drawn by a true syllogism. What usually gets left out is only the complete statement of the premises and the conclusion. The reader generally knows one premise or the other already, or can easily guess what is being assumed, and it is tedious, even irritating, to spell out at full length what the reader already knows. Suppose I say, "atmospheric air must have weight because it is a material substance." I am certainly using a syllogism. But I think it quite unnecessary to state the premise whose truth I am plainly assuming, namely "whatever is a material substance has weight." The conclusion of the syllogism is the first proposition, "atmospheric air has weight." The middle term is "material substance," which does not appear in the conclusion. The minor term is "atmospheric air," and the major term is "having weight." The complete syllogism is plainly this:

	All material substances have weight,
	Atmospheric air is a material substance;
	Therefore atmospheric air has weight.

This is in the very common and useful mood Barbara.

A syllogism that is stated incompletely is usually called an enthymeme. The name is often supposed to come from two Greek words, ἐν (en), "in," and θυμός (thymos), "mind." On that reading it means that some piece of knowledge is held in the mind and supplied as a tacit premise, that is, a silent or understood one. Most often the missing premise is the major premise, and then the enthymeme can be said to be of the First Order. Less often it is the minor premise that goes unstated, and then the enthymeme is of the Second Order. Here is an argument of that kind: "Comets must be subject to the law of gravitation; for this is true of all bodies which move in elliptical orbits." It is so obviously implied that comets move in elliptical orbits that it would be tedious to state it as the minor premise of a complete syllogism in the mood Barbara, like this:

	All bodies moving in elliptical orbits are subject to the law of gravitation;
	Comets move in elliptical orbits;
	Therefore comets are subject to the law of gravitation.

Occasionally it is the conclusion of a syllogism that is left unstated, and then the enthymeme can be said to belong to the Third Order. This happens with epigrams and other witty sayings, where the whole wit often lies in making an unstated truth obvious. Sir William Hamilton gives, as an example of this kind of enthymeme, the famous epigram that the English scholar Porson wrote about a German scholar of his own day, Gottfried Hermann:

	"The Germans in Greek
	Are sadly to seek;
	Not five in five score,
	But ninety-five more;
	All, save only Hermann,
	And Hermann's a German."

The writer pretends to make an exception of Hermann. But he cleverly slips in the suggestion that, since Hermann is a German, he does not really know Greek either. In the same way, the wonderful speech Antony makes over Caesar's body, in Julius Caesar, Shakespeare's greatest historical play, contains a whole series of syllogistic arguments whose conclusions are only hinted at.

Even a single proposition can have the force of a syllogism, if it clearly brings to mind a second premise from which a conclusion can then be drawn. A saying of the radical writer John Horne Tooke, "Men who have no rights cannot justly complain of any wrongs," seems to be a case in point. Very few people have never felt wronged at one time or another, and so they would be likely to argue, whether their premises were true or false, like this:

	Men who have no rights cannot justly complain of any wrongs;
	We can justly complain;
	Therefore we are not men who have no rights.

In other words, we have rights.

Syllogisms can be joined and combined in various ways, and it is convenient to have special names for the different parts of a complex argument. A syllogism that proves, or gives a reason for, one of the premises of another syllogism is called a Prosyllogism. A syllogism that has as one of its premises the conclusion of another syllogism is called an Episyllogism.

Take this example:

	All B's are A's,
	And all C's are B's;
	Therefore all C's are A's.
	But all D's are C's;
	Therefore all D's are A's.

This plainly contains two syllogisms in the mood Barbara. The first is a Prosyllogism with respect to the second, and the second is an Episyllogism with respect to the first.

The special name Epicheirema is given to a syllogism in which either premise is proved or supported by a reason, so that a prosyllogism is implied but only partly expressed. So the form

	All B's are A's, for they are P's,
	And all C's are B's, for they are Q's;
	Therefore all C's are A's,

is a double Epicheirema, since it gives reasons for both premises. The reader will easily break it down into three complete syllogisms in the mood Barbara.

A more interesting form of reasoning is the chain of syllogisms usually called the Sorites, from the Greek word σωρός (sōros), meaning a heap. It is usually stated like this:

	All A's are B's,
	All B's are C's,
	All C's are D's,
	All D's are E's;
	Therefore all A's are E's.

The chain can be carried on to any length, as long as it is perfectly continuous, so that every term except the first and the last appears twice, once as a subject and once as a predicate. It hardly needs pointing out that a sorites really contains a series of syllogisms stated incompletely, like this:

Each syllogism supplies a premise to the one after it, and so it is the prosyllogism of that one. Equally, any syllogism in the chain can be regarded as the episyllogism of the one before it.

In the sorites above, all the premises were universal and affirmative. But a sorites may contain one particular premise, provided it is the first, and one negative premise, provided it is the last. The reader can easily check by trying it that if any premise except the first were particular, the fallacy of undistributed middle would be committed. That is because one of the middle terms would then be the predicate of one affirmative premise and the subject of another, particular, premise. And if any premise except the last were negative, there would be a fallacy of illicit process of the major term.

We should not suppose that the forms of the syllogism described so far are the only kinds of reasoning actually used in science or in everyday life. Besides the hypothetical and disjunctive syllogisms, and some other forms to be described in later lessons, there are in fact many ways of reasoning that logicians have so far paid little attention to. This was clearly pointed out more than two hundred years ago by the writers of the Port Royal Logic. The book was first printed in 1662, but it has been reprinted many times since and translated into a great many languages. It takes its name from Port Royal, a place near Paris where a small religious community lived. The authors of the book, Arnauld and Nicole, and one contributor to it, the great philosopher and mathematician Pascal, were the most famous members of that community. The Port Royal Logic was to a large extent the basis of the well-known Watts's Logic, but the reader can now be referred to an excellent translation of the original work by Professor Spencer Baynes of St Andrews.

Many improvements to logic can be found in this book, such as the doctrine of extension and intension explained in Lesson 5. Moreover, in the 9th Chapter of the 3rd Part it wisely points out that "little pains are taken in applying the rules of the syllogism to reasonings of which the propositions are complex, though this is often very difficult, and there are many arguments of this nature which appear bad, but which are nevertheless very good; and besides, the use of such reasonings is much more frequent than that of syllogisms which are quite simple." The book gives some examples of the complex syllogisms it means, such as this one:

	The sun is a thing without feeling,
	The Persians worship the sun;
	Therefore the Persians worship a thing without feeling.

This argument cannot be proved by the rules of the syllogism. Yet it is not only plainly true, it is an extremely common kind of argument. Here is another example:

	The Divine Law commands us to honor kings;
	Louis XIV is a king;
	Therefore the Divine Law commands us to honor Louis XIV.

The reader will also find arguments that are perfectly valid and syllogistic, but are worded so that they seem to have four different terms, and so to break one of the rules of the syllogism. Suppose I say, "Diamonds are combustible, for they are composed of carbon and carbon is combustible." Four terms are used here: diamonds, combustible, composed of carbon, and carbon. But it is easy to rearrange the propositions so as to get a simple syllogism without really changing the sense, and then we have:

	What is composed of carbon is combustible;
	Diamonds are composed of carbon;
	Therefore diamonds are combustible.

Among the Miscellaneous Examples at the end of Lesson 21 are some short arguments, taken from Bacon's Essays and other writings, which the student can reduce to syllogistic form by easy changes of wording. But it should be clearly understood that changes like these are not part of logic itself. They belong more properly to the science of language.

I should explain here that the syllogism and the sorites can be expressed either in the order of extension or in the order of intension. In terms of the number of individual things, the noble metals are part of the metals, and the metals are part of the elements. But in terms of intension, that is, the qualities the names imply, element is part of metal, and metal is part of noble metal. Again, in extension the plant genus Anemone is part of the order Ranunculaceae (the buttercup family), and that order is part of the great class of Exogens (roughly, the dicots). But in intension the character of Exogen is part of the character of Ranunculaceae, and that is part of the character of Anemone. Syllogistic reasoning is just as valid and just as obvious either way, and we could show the two modes in ordinary language like this:

	Extensive Syllogism.
	All Ranunculaceae are Exogens;
	The Anemone is one of the Ranunculaceae;
	Therefore the Anemone is an Exogen.
	Intensive Syllogism.
	All the qualities of Ranunculaceae are qualities of Anemone;
	All the qualities of Exogen are qualities of Ranunculaceae;
	Therefore all the qualities of Exogen are qualities of Anemone.

Any sorites can be shown in the same way, in either extension or intension.

On the Aristotelian doctrine of the Enthymeme, see Mansel's Aldrich, App. Note F, and Hamilton's Lectures on Logic, Lecture XX. Port Royal Logic, translated by T. Spencer Baynes, 7th ed., Edinburgh.

Questions and Exercises

1. Explain the meaning of each of these terms: Enthymeme, Prosyllogism, Episyllogism, Epicheirema, Sorites.

2. Make up an example of a syllogism that has two prosyllogisms.

3. Construct a sorites with four premises, and break it down into separate syllogisms.

4. What rules must a sorites obey?

5. The reader is asked to analyze the following arguments, to pick out the ones that are false, and to find which rules of the syllogism they break. If an argument seems valid, the reader should find its figure and mood, state it in correct logical form, and then, if it is in an imperfect figure, prove it by reducing it to the first figure. The first six examples should be arranged in both the extensive and the intensive orders.

1. None but mortals are human beings. Monarchs are human beings. Therefore monarchs are mortals.

2. Personal deformity is an affliction of nature. Disgrace is not an affliction of nature. Therefore personal deformity is not disgrace.

3. Some statesmen are also authors; for such are Mr. Gladstone, Lord Derby, Lord Russell, and Sir G. C. Lewis.

4. This explosion must have been caused by gunpowder; for nothing else would have had enough force.

5. Every person should be moderate; for excess will cause disease.

6. Blessed are the merciful; for they shall obtain mercy.

7. Since almost all the organs of the body have a known use, the spleen must have some use.

8. Cogito, ergo sum. (I think, therefore I exist.)

9. Some speculative people are unworthy of trust; for they are unwise, and no unwise person can be trusted.

10. No idle person can be a successful writer of history; therefore Hume, Macaulay, Hallam and Grote must have been hard-working.

11. Whoever spares the rod hates his child; therefore the parent who loves his child does not spare the rod.

12. Comets must consist of heavy matter; for otherwise they would not obey the law of gravitation.

13. Lithium is an element; for it is an alkali-producing substance, which is a metal, which is an element.

14. Rational beings are accountable for their actions; brute animals, not being rational, are therefore exempt from responsibility.

15. A singular proposition is a universal one; for it applies to the whole of its subject.

16. Whatever tends to draw the mind away from low pursuits deserves to be encouraged; classical learning does this, since it gives us a taste for intellectual pleasures; therefore it deserves to be encouraged.

17. Bacon was a great lawyer and statesman; and since he was also a philosopher, we may infer that any philosopher may be a great lawyer and statesman.

18. Immoral companions should be avoided; but some immoral companions are intelligent people, so some intelligent people should be avoided.

19. Mathematical study undoubtedly improves the reasoning powers; but since the study of logic is not mathematical study, we may infer that it does not improve the reasoning powers.

20. Every fair-minded man acknowledges merit in a rival; every learned man does not do so; therefore every learned man is not fair-minded.

Lesson 19: Of Conditional Arguments

You will remember that when we dealt with propositions we divided them into two distinct kinds: Categorical Propositions and Conditional Propositions. So far we have looked only at the first kind. Now we must go on to describe Conditional propositions, and the arguments that can be built out of them.

Logicians have usually described a Conditional proposition as two or more Categorical propositions joined by a conjunction. They can be joined in two ways, and this gives two very different species of conditional, which we will call Hypothetical Propositions and Disjunctive Propositions. The diagram below shows how the different kinds of propositions are related:

We can also describe a conditional proposition as one that makes a statement under a certain condition, or qualification, which restricts how far the statement applies. In the hypothetical form the condition is brought in by the conjunction if, or by some other word that does the same job. So

	"If iron is impure, it is brittle"

is a hypothetical proposition made up of two distinct categorical propositions. The first, "Iron is impure," is called the Antecedent. The second, "It is brittle," is called the Consequent. Here "impurity" is the condition, or qualification, that limits the application of the predicate brittle to iron. In his celebrated book The Diversions of Purley, Horne Tooke claimed that all conjunctions are the remains, or worn-down forms, of verbs. This is certainly true of the hypothetical conjunction. In old English the word if is written gif or gyf, and it undoubtedly comes from the verb to give. Even today we can put in its place any verb of similar meaning, such as grant, allow, or suppose. So we can say:

	"Grant that iron is impure, and it is brittle."
	"Supposing that iron is impure, it is brittle."

The hypothetical proposition could be used in arguments of various forms, but only two of these are important enough to have been given special names. The hypothetical syllogism has two premises, called the major and the minor, just like the ordinary syllogism. The major premise is hypothetical in form. The minor premise is categorical, and depending on whether it is affirmative or negative, the argument is called a Constructive or a Destructive hypothetical syllogism. So the form

	If A is B, C is D;
	But A is B;
	Therefore C is D,

is a constructive hypothetical syllogism.

Notice carefully that the minor premise affirms the antecedent of the major premise. That is why the argument is said to be in the modus ponens, the mood that posits, or affirms. It is probably one of the most familiar and common kinds of argument there is. The form

	If A is B, C is D;
	But C is not D;
	Therefore A is not B,

is the matching Destructive hypothetical syllogism, also called the modus tollens, the mood that removes the consequent. Notice carefully, again, that it is the consequent, not the antecedent, that is denied.

Only one rule is needed for testing whether such syllogisms are valid, and it sums up what we have just seen: either the antecedent must be affirmed, or the consequent denied. Break this rule in either direction and you commit a serious fallacy. Take this apparent argument:

	If A is B, C is D;
	But C is D;
	Therefore A is B.

It is really a fallacy, which we can call the fallacy of affirming the consequent. It is easy to see why it fails. Nothing in the premise says that "A being B" is the only condition under which C is D. C might be D when E is F, or when G is H, or in a hundred other circumstances. So the mere fact that C is D is not enough to prove that A is B. For example, if a man has an avaricious, or greedy, character, he will refuse to give money for useful purposes. But it does not follow that everyone who refuses to give money for such purposes is avaricious. There may be many good reasons or motives for refusing. He may have no money; he may think the purpose is not a useful one; or he may have more useful purposes in mind.

A matching fallacy comes from denying the antecedent, as in this form:

	If A is B, C is D;
	But A is not B;
	Therefore C is not D.

The mistake can be explained the same way. Nothing says that "A being B" is the only condition of C being D. So we may deny that this one condition holds, and the consequent may still happen to be true for other reasons we know nothing about. If a man is not avaricious, then, we cannot conclude that he is sure to give money whenever he is asked. Or take this example:

"If the study of Logic furnished the mind with a multitude of useful facts, like the study of other sciences, it would deserve cultivation; but it does not furnish the mind with a multitude of useful facts; therefore it does not deserve cultivation."

This argument is plainly fallacious, because picking up a multitude of useful facts is not the only reason for recommending the study of a science. The reason Logic deserves to be cultivated is that it corrects and exercises our powers of judgment and reasoning. The argument above ignores that this other purpose exists, and it plainly commits the fallacy of denying the antecedent.

Books on logic usually describe the hypothetical proposition and syllogism as if they were different in nature from the categorical proposition and syllogism. But it has long been known that hypotheticals can be reduced to the categorical form and brought under the ordinary rules of the syllogism. As a general rule, a hypothetical proposition can easily be turned into a universal affirmative proposition (A) with exactly the same meaning. Our example, "If iron is impure, it is brittle," simply becomes "Impure iron is brittle." When we make this change in a hypothetical syllogism, we find we have to supply a new minor term. Take the case

	If iron is impure it is brittle;
	But it is impure;
	Therefore it is brittle.

Here we have to put "the iron in question" in place of the indefinite pronoun it, and we get a correct categorical syllogism in the mood Barbara:

	Impure iron is brittle;
	The iron in question is impure iron;
	Therefore the iron in question is brittle.

Sometimes the reduction needs a bigger change of wording. For instance,

	If the barometer is falling, bad weather is coming;
	But the barometer is falling;
	Therefore bad weather is coming,

can be put in this form:

	The circumstances of the barometer falling are the circumstances of bad weather coming;
	But these are the circumstances of the barometer falling;
	Therefore these are the circumstances of bad weather coming.

As an example of the Destructive Hypothetical syllogism we can take:

	If Aristotle is right, slavery is a proper form of society;
	But slavery is not a proper form of society;
	Therefore Aristotle is not right.

As a categorical, this becomes:

	The case of Aristotle being right is the case of slavery being a proper form of society;
	But this is not the case of slavery being a proper form of society;
	Therefore this is not the case of Aristotle being right.

When no other way of wording will reduce a hypothetical, the words "case of" always will.

Now we can easily show that the fallacy of affirming the consequent is really a breach of the third rule of the syllogism, and leads to an undistributed middle term. We can use the same example as before:

	If a man is avaricious he will refuse money;
	But he does refuse money;
	Therefore he is avaricious.

As a categorical syllogism this becomes:

	All avaricious men refuse money;
	But this man refuses money;
	Therefore this man is avaricious.

This is the mood AAA in the second figure. The middle term, refusing money, is undistributed in both premises, so the argument is entirely fallacious.

In the same way, the fallacy of denying the antecedent amounts to the illicit process of the major. Our earlier example (the one about the study of Logic, given earlier in this lesson) can be put like this:

"A science which furnishes the mind with a multitude of useful facts deserves cultivation; but Logic is not such a science; therefore Logic does not deserve cultivation."

This apparent syllogism is in the mood AEE in the first figure. It breaks the fourth rule of the syllogism, because the major term, deserving cultivation, is distributed in the negative conclusion but not in the affirmative major premise.

We now move on to the disjunctive proposition. Instead of a single predicate, it has several alternatives joined by the disjunctive conjunction or, and any one of them may be affirmed of the subject. "A member of the House of Commons is either a representative of a county, or of a borough, or of a University" is an example of such a proposition. It has three alternatives, but there can be any number, from two upward.

The disjunctive syllogism has a disjunctive major premise, with a categorical proposition, either affirmative or negative, as its minor premise. That gives two moods. The affirmative one is called by the Latin name modus ponendo tollens (the mood that denies by affirming), and it can be stated like this:

	A is either B or C,
	But A is B;
	Therefore A is not C.

This form of argument assumes that if one alternative of a disjunctive proposition is held to be true, the others cannot be true as well. Thus "the time of year must be either spring, summer, autumn or winter," and if it is spring, it cannot be summer, autumn or winter, and so on. But Whately, Mansel and Mill, along with many earlier logicians, have objected that this does not always hold. Suppose we say that "a good book is valued either for the usefulness of its contents or the excellence of its style." It certainly does not follow that because the contents of a book are useful, its style is not excellent. We generally choose alternatives that are inconsistent with each other, but logic does not require it.

The other form of disjunctive syllogism, called the modus tollendo ponens (the mood that affirms by denying), is always, necessarily, conclusive. It goes like this:

	A is either B or C,
	But A is not B;
	Therefore A is C.

So if we suppose that a book is valued only for the usefulness of its contents or for the excellence of its style, it follows that if a book is valued, but not for the first reason, it must be valued for the second; and the other way around. If the time of year is not spring, it must be summer, autumn or winter. If it is neither autumn nor winter, it must be either spring or summer, and so on. In short, if any of the alternatives are denied, the rest remain to be affirmed as before. Notice that the disjunctive syllogism is governed by completely different rules from the ordinary categorical syllogism. In the disjunctive syllogism a negative premise gives an affirmative conclusion; in the categorical syllogism it gives a negative one.

One more form of argument remains. It is called the Dilemma, because it starts by assuming two alternatives, usually called the horns of the dilemma, and still proves something whichever one holds (from the Greek δι- (di-), two, and λῆμμα (lēmma), assumption). Mr. Mansel defines this argument as "a syllogism, having a conditional major premise with more than one antecedent, and a disjunctive minor." It can be stated in at least three forms. The first is called the Simple Constructive Dilemma:

	If A is B, C is D; and if E is F, C is D;
	But either A is B, or E is F;
	Therefore C is D.

For example: "if a science furnishes useful facts, it is worth cultivating; and if studying it exercises the reasoning powers, it is worth cultivating; but either a science furnishes useful facts, or studying it exercises the reasoning powers; therefore it is worth cultivating."

The second form is the Complex Constructive Dilemma, which goes like this:

	If A is B, C is D; and if E is F, G is H;
	But either A is B, or E is F;
	Therefore either C is D, or G is H.

It is called complex because the conclusion is in disjunctive form. As an example we can take this argument: "If a statesman who sees that his former opinions were wrong does not change his course, he is guilty of deceit; and if he does change his course, he is open to a charge of inconsistency; but either he does not change his course or he does; therefore he is either guilty of deceit, or open to a charge of inconsistency." In this case, as in most dilemmas, the terms A, B, C, D and so on are not all different.

The Destructive Dilemma is always complex, because otherwise it could be broken up into two unconnected destructive hypothetical syllogisms. Its form is:

	If A is B, C is D; and if E is F, G is H;
	But either C is not D, or G is not H;
	Therefore either A is not B, or E is not F.

For instance: "If this man were wise, he would not speak irreverently of Scripture in jest; and if he were good, he would not do so in earnest; but he does it either in jest or in earnest; therefore he is either not wise, or not good."

Footnote: Whately.

Still, dilemmatic arguments are more often fallacious than not. It is rarely possible to find cases where two alternatives cover every possibility, unless one of them is simply the negative of the other, in keeping with the law of excluded middle (see Lesson 14). Suppose we argued that "if a pupil is fond of learning he needs no stimulus, and if he dislikes learning no stimulus will be of any use; but since he is either fond of learning or dislikes it, a stimulus is either needless or of no use." We would plainly be assuming the disjunctive minor premise without any right to it. Fondness and dislike are not the only two possible alternatives. Some pupils are neither fond of learning nor dislike it, and for them a stimulus in the shape of rewards may be a good thing. You can prove almost anything if you are allowed to pick out two of the possible alternatives that suit you and argue from those alone.

A dilemma can often be retorted, that is, rebutted by producing an equally conclusive dilemma that proves the opposite. According to Aristotle, an Athenian mother spoke to her son like this: "Do not go into public life; for if you say what is just, men will hate you; and if you say what is unjust, the gods will hate you." Aristotle suggests the following retort: "I ought to go into public life; for if I say what is just, the gods will love me; and if I say what is unjust, men will love me."

Footnote: Mansel's Aldrich, Appendix, Note I, on the Hypothetical Syllogism.

Questions and Exercises

1. What are the kinds of conditional propositions, and by what signs can you recognize them?

2. What are the rules of the hypothetical syllogism?

3. Which categorical fallacies do breaches of these rules correspond to?

4. Pick out those of the following that are valid arguments, and reduce them to the categorical form; explain what is fallacious in the reasoning of the others.

(1) Rain has fallen if the ground is wet; but the ground is not wet; therefore rain has not fallen.

(2) If rain has fallen, the ground is wet; but rain has not fallen; therefore the ground is not wet.

(3) The ground is wet, if rain has fallen; the ground is wet; therefore rain has fallen.

(4) If the ground is wet, rain has fallen; but rain has fallen; therefore the ground is wet. Note: in these, as in other logical examples, the student must argue only from the premises, and not from any other knowledge of the subject matter.

5. Show that the canons of the syllogism (see Lesson 14) can be stated equally well in the hypothetical or the categorical form.

6. State the following as a Disjunctive or a Dilemmatic argument, and name the kind it belongs to.

If pain is severe it will be brief; and if it lasts long it will be slight; therefore it is to be patiently borne.

Lesson 20: Logical Fallacies

To get a satisfactory knowledge of the rules of correct thinking, we need to become familiar with the most common kinds of fallacy, that is, the ways in which we often fall into bad reasoning by neglecting the rules of logic. In the earlier lessons we have been learning, so to speak, how to find the right road. Our job here is to find the turnings where we are most likely to take the wrong one.

In describing the fallacies I will follow the order and the way of classifying them that has been usual for more than 2,000 years, in fact ever since the great teacher Aristotle first explained the fallacies. Under this arrangement the fallacies are divided into two main groups: the logical fallacies and the material fallacies.

1. The logical fallacies are the ones that lie in the mere form of the statement, or, as the old Latin expressions put it, in dictione or in voce (in the wording, or in the words). So it is supposed that fallacies of this kind can be discovered without any knowledge of the subject-matter the argument is about.

2. The material fallacies, on the other hand, arise outside the mere verbal statement, or, as it is said, extra dictionem (outside the wording). So they are concerned with the subject of the argument, in re (in the matter), and they can only be detected and set right by people who know the subject.

The first group, the logical fallacies, can be divided further into the purely logical and the semi-logical. In the purely logical class we can include the plain breaches of the rules of the syllogism, which have already been described. So we can list the Purely Logical Fallacies as:

1. Fallacy of four terms (Quaternio Terminorum): a breach of Rule 1;

2. Fallacy of undistributed middle: a breach of Rule 3;

3. Fallacy of illicit process, of the major or the minor term: a breach of Rule 4;

4. Fallacy of negative premises: a breach of Rule 5; and also breaches of the 6th rule, which have been given no separate name. Breaches of the 7th and 8th rules can be resolved into the ones above (see Lesson 17), but they can also be described as they were at the end of Lesson 15.

The other part of the class of logical fallacies contains the Semi-logical fallacies. There are six of them:

	1. Fallacy of Equivocation.
	2. Fallacy of Amphiboly.
	3. Fallacy of Composition.
	4. Fallacy of Division.
	5. Fallacy of Accent.
	6. Fallacy of Figure of Speech.

I will describe and illustrate these one after another.

Equivocation is using the same term in two different senses. Any of the three terms of the syllogism can suffer from this fallacy, but usually it is the middle term, used in one sense in one premise and in another sense in the other. In that case it is often called the fallacy of ambiguous middle. When we separate the two meanings by expressing each of them in other suitable words, it becomes clear that the supposed syllogism contains four terms. So the fallacy of equivocation can be considered a disguised fallacy of four terms. Suppose someone argued that "all criminal actions ought to be punished by law; prosecutions for theft are criminal actions; therefore prosecutions for theft ought to be punished by law." In the first premise a "criminal action" is a crime. In the second it is the legal name for a criminal case brought in court. It is quite obvious that the term means totally different things in the two premises, and that there is no true middle term at all. Often, though, the ambiguity is subtle and difficult, so that people can hold different opinions about it. For example, we might argue:

"He who harms another should be punished. He who communicates an infectious disease to another person harms him. Therefore he who communicates an infectious disease to another person should be punished."

Whether this counts as a correct argument depends on what kinds of actions we think come under the term harm: whether we think carelessness, or only malice, is needed to make something harm. Many difficult legal questions are of this kind. For instance:

	Nuisances are punishable by law;
	To keep a noisy dog is a nuisance;
	To keep a noisy dog is punishable by law.

The question here would turn on how great a nuisance has to be before the law will step in to prevent it. Or again:

	Interference with another man's business is illegal;
	Underselling interferes with another man's business;
	Therefore underselling is illegal.

Here the question turns on the kind of interference, and it is obvious that underselling (selling more cheaply than a competitor) is not the kind of interference the major premise is talking about.

The Fallacy of Amphiboly, or amphibology, is an ambiguous grammatical structure in a sentence, which leads to a misunderstanding. A famous example comes in the prophecy of the spirit in Shakespeare's Henry VI: "The Duke yet lives that Henry shall depose." In plain words, "The Duke is still alive that Henry shall depose." The grammar leaves it completely uncertain whether the Duke will depose Henry or Henry will depose the Duke. This prophecy is no doubt an imitation of the ones the ancient oracle of Delphi is said to have given, and it seems that this fallacy was a great resource for oracles who were not confident in their own powers of foresight. The Latin language gives plenty of room for misreadings, because it does not require any fixed order for the words of a sentence. When there are two nouns in the accusative case with an infinitive verb, it may be hard to tell, except from the context, which of them is meant to come before the verb in sense, that is, which one is doing the action. The double meaning that can be given to "twice two and three" comes from amphiboly. It can be 7 or 10, depending on whether we add the 3 after multiplying or before. When sentences are carelessly built, it is often impossible to tell which part an adverb or a qualifying phrase belongs to. If someone says "I finished my business and returned the day after," the business may have been finished on the day after as well as the return. But it could just as well have been finished on the day before. An ambiguity of this kind can usually be avoided by simply changing the order of the words, for instance: "I finished my business, and, on the day after, returned." Amphiboly can also arise from mixing up the subjects and predicates in a compound sentence. Take "platinum and iron are very rare and useful metals." It would be amphiboly if I meant the predicate useful to go with platinum and rare with iron, which is not what the sentence intends. The word "respectively" is often used to show the reader that he is not free to apply every predicate to every subject: each predicate goes with its own subject, in order.

The Fallacy of Composition is a special case of equivocation. It comes from confusing a universal term with a collective term. In the premises of a syllogism we may say something about a class of things distributively, that is, about each one of them separately, and then in the conclusion infer the same thing about the whole put together. We can say, for example, that "all the angles of a triangle are less than two right angles," meaning that any one of the angles is less than two right angles. But we must not infer that all the angles put together are less than two right angles. We must not argue that because every member of a jury is very likely to judge wrongly, the jury as a whole is also very likely to judge wrongly. Nor must we argue that because each of the witnesses in a court case may give false or mistaken evidence, no confidence can be placed in the agreeing testimony of a number of witnesses. It is through a fallacy of Composition that protective duties (tariffs that shield a home industry from foreign competition) are still sometimes defended. Because any one trade, or any few trades, that enjoy protective duties are benefited by them, people suppose that all trades at once could be benefited in the same way. But this is impossible, because protecting one trade raises prices, and so injures all the others.

The Fallacy of Division is the reverse of the one before. It consists in using the middle term collectively in the major premise but distributively in the minor, so that the whole is divided into its parts. For example, someone might argue: "All the angles of a triangle are (together) equal to two right angles; ABC is an angle of a triangle; therefore ABC is equal to two right angles." Or again: "The inhabitants of the town consist of men, women and children of all ages; those who met in the Guildhall (the town hall) were inhabitants of the town; therefore they consisted of men, women and children of all ages." Or: "The judges of the court of appeal cannot misinterpret the law; Lord A.B. is a judge of the court of appeal; therefore he cannot misinterpret the law."

The Fallacy of Accent is any ambiguity that comes from a misplaced accent, or emphasis, on some word of a sentence. An absurd example is liable to happen when reading 1 Kings 13:27, where it is said of the prophet: "And he spake to his sons, saying, Saddle me the ass. And they saddled him." ("Saddle me the ass" means "saddle the donkey for me.") The word him is printed in italics because the translators of the Authorized Version (the King James Bible) supplied it themselves. It is meant to refer to the ass, but read with a stress on it, it suggests a very different meaning: that the sons saddled their father. The Commandment "Thou shalt not bear false witness against thy neighbor" can be made, by a slight emphasis of the voice on the last word, to imply that we are free to bear false witness against other people. Mr. De Morgan, who points this out, also notes that misquoting an author, by unfairly separating a word from its context, or by italicizing words that were never meant to be italicized, produces cases of this fallacy.

It is curious to see how many different meanings the same sentence can be given, depending on which word we put the emphasis on. Take the sentence "The study of logic is not supposed to give us a knowledge of many useful facts." It can be made to imply that the study of logic does give such knowledge, even though it is not supposed to; or that it gives a knowledge of a few useful facts; or that it gives a knowledge of many useless facts. We can explain this ambiguity as follows. If you deny that a thing has the group of qualities A, B, C, D, your statement is true as long as any one of those qualities is missing, and an accent in speaking is often used to point out the one the speaker believes is missing. If you deny that a particular fruit is ripe and sweet and well-flavored, it may be unripe and sweet and well-flavored; or ripe and sour and well-flavored; or ripe and sweet and badly flavored; or any two, or even all three, of the qualities may be missing. But if you deny that it is ripe and sweet and well-flavored, your denial would be understood to refer to the last quality. Jeremy Bentham was so afraid of being misled by this fallacy of accent that, as I have heard, he employed a person to read to him who had a peculiarly monotonous way of reading.

The Fallacy of the Figure of Speech is the sixth and last of the semi-logical fallacies, and it is a very trivial one. It seems to consist in any grammatical mistake, or any confusion between one part of speech and another. Aristotle quite seriously gives this example: "Whatever a man walks he tramples on; a man walks the whole day; therefore he tramples on the day." Here an adverbial phrase ("the whole day," meaning for how long he walks) is turned into the object of the verb, as if the day were something he walks, the way he walks the streets.

Lesson 21: Material Fallacies

We come now to the Material fallacies. They matter a great deal, although they are not easy to illustrate with short examples. Aristotle listed seven kinds, and later logicians have kept his list:

1. The Fallacy of Accident.

2. The Converse Fallacy of Accident.

3. The Irrelevant Conclusion.

4. The Petitio Principii.

5. The Fallacy of the Consequent, or Non sequitur.

6. The False Cause.

7. The Fallacy of Many Questions.

The first two are most easily described together. The fallacy of accident is the mistake of arguing from a general rule to a special case where some accidental circumstance makes the rule not apply. The converse fallacy is the mistake of arguing from a special case to a general rule. This second fallacy is usually described by the Latin phrase a dicto secundum quid ad dictum simpliciter, which means "from a statement made under a condition to a statement made simply, without that condition." Professor De Morgan, in his very interesting Chapter on Fallacies, has pointed out that we ought to add a third fallacy of this kind: arguing from one special case to another special case.

Footnote: Formal Logic, Chapter XIII.

I will try to illustrate these kinds of fallacy with a few examples. Be warned, though: it is often very hard to say which of the three a particular example belongs to. Here is a very ancient example, repeated in almost every logic handbook: "What you bought yesterday you eat today; you bought raw meat yesterday; therefore you eat raw meat today." The conclusion makes its claim about meat with the accidental quality of rawness added. But the first premise is plainly talking about the meat as a substance, without regard to its accidental condition. So this is a case of the direct fallacy. If, on the other hand, someone argues that because wine acts as a poison when taken in excess it is always a poison, we have the converse fallacy.

Here is another case of the direct fallacy of accident: to infer that a magistrate is justified in using his power to advance his own religious views, because every man has a right to teach his own opinions. Clearly a magistrate, as a man, has the same rights as other men. But in his capacity as a magistrate he is set apart from other men, and he must not claim for his special powers what is true only of his rights as a man. For another instance, take this: "Anyone who thrusts a knife into another person should be punished; a surgeon does this when he operates; therefore he should be punished." The fallacy here is absurdly obvious, but it is not so obvious how we should classify the error. We might say that, as a general rule, whoever stabs or cuts another person is to be punished, unless it can be shown that it was done under exceptional circumstances, as by a properly qualified surgeon acting for the patient's good. Read that way, the example belongs to the direct fallacy of accident. Read another way, the first premise might mean only the special case of thrusting a knife into someone out of malice. To argue from that case to the case of a surgeon would then be to infer from one special case to another special case.

It is certainly true that giving to beggars encourages begging and does harm. But if we take this to mean that help should never be given to anyone who asks for it, we fall into the converse fallacy of accident: we infer about everyone who asks for charity what is true only of those who beg for a living. In the same way, it is a very good rule to avoid lawsuits and quarrels, but only as a general rule, because circumstances often arise in which going to law is a plain duty. Nearly all the difficulties we meet in matters of law and moral duty come from this: we can never be sure exactly which cases a legal or moral rule covers and which it does not. That is why there are endless differences of opinion, even among the judges of the land.

The Third Material Fallacy is the Irrelevant Conclusion, technically called the Ignoratio Elenchi, which means literally Ignorance of the Refutation. It consists in arguing to the wrong point: proving one thing in such a way that people suppose something else has been proved. Here again it is hard to give short examples, because this fallacy usually turns up in the course of long speeches, where the flood of words and figures leaves room for muddled thinking and forgetfulness. In fact, this fallacy is the great standby of anyone who has to defend a weak case. It is not unknown in the legal profession. The story goes that a solicitor for the defendant in a lawsuit once handed the barrister (the lawyer who would argue the case in court) his brief marked: "No case; abuse the plaintiff's attorney." Anyone who argues like this is using what is called the argumentum ad hominem, an argument that rests not on the merits of the case but on the character or position of the people involved in it, and he commits this fallacy. If a man is accused of a crime, it is no answer to say that the prosecutor is just as bad. If a great change in the law is proposed in Parliament, it is an Irrelevant Conclusion to argue that the person proposing it is not the right man to bring it forward. Everyone who gives advice lays himself open to the retort that he should practice what he preaches, or that people who live in glass houses should not throw stones. Even so, there is no necessary connection between the character of the person giving advice and the quality of the advice.

The argumentum ad populum is another form of Irrelevant Conclusion. It consists in addressing to a crowd arguments designed to stir up their feelings and keep them from judging the matter in hand calmly. It is the great weapon of orators and demagogues.

Petitio Principii is a familiar name, and the fallacy it names is exactly described by the phrase begging the question. Another good name for it is circulus in probando, "a circle in the proof." It consists in using the conclusion itself as one of the premises of the argument. Of course, the conclusion of a syllogism must always be contained in, or implied by, the premises. But it is contained in them only when the premises are put together, and the premises must be assertions clearly different from the conclusion. Take this syllogism:

	B is C,
	A is B,
	therefore A is C.

Here the conclusion is proved by being deduced from two propositions, neither of which is the same as the conclusion. But suppose the truth of one of these premises itself depends on the following syllogism:

	C is B,
	A is C,
	therefore A is B.

Then plainly we are trying to prove a proposition by means of itself, which makes as much sense as trying to hold a body up by itself. This kind of fallacy is not easy to illustrate with examples, because it usually happens in long arguments, and especially in wordy metaphysical writing. We are, however, very likely to fall into it when we mix plain Anglo-Saxon words with Latin or Greek ones, so that we seem to prove one proposition by another which is really the same proposition in different words. For example: "Consciousness must be immediate cognition of an object; for I cannot be said really to know a thing unless my mind has been affected by the thing itself."

This fallacy is especially likely to happen when we use the disjunctive syllogism. If we list only the alternatives that favor one view and forget the others, it is easy to prove anything at all. There is an instance of this in the famous sophism by which some of the ancient Greek philosophers proved that motion was impossible. A moving body, they said, must move either in the place where it is or in the place where it is not. Now it is absurd to say that a body can be where it is not; and if it moves, it cannot be in the place where it is. Therefore it cannot move at all. The error lies in assuming a premise that begs the question. The fact, of course, is that the body moves between the place where it is at one moment and the place where it is at the next.

Jeremy Bentham pointed out, however, that even a single name can carry a Petitio Principii with it. Suppose a church assembly, or synod, is discussing whether a certain doctrine ought to be condemned. It would be a Petitio Principii to argue that the doctrine is heresy and therefore ought to be condemned. To assert that it is heresy is to beg the question, because everyone understands by heresy a doctrine that is to be condemned. In the same way, in Parliament a bill is often opposed on the ground that it is unconstitutional and therefore ought to be rejected. But since no one can give a precise definition of what is or is not constitutional, this means little more than that the opponent dislikes the measure. Bentham gave names used in this fallacious way a good title: Question-begging Epithets. We beg the question in just the same way when we oppose a change by saying that it is un-English.

The Fallacy of the Consequent is better known by the familiar phrase non sequitur ("it does not follow"). We may give this name to any argument so loose and disconnected that no one can find any force in it at all. It amounts to little more than asserting a conclusion that has no connection with the premises. Professor De Morgan gives this example (episcopacy means the government of a church by bishops): "Episcopacy is of Scripture origin; the Church of England is the only episcopal Church in England; ergo, the Church established is the Church that should be supported."

By the Fallacy of the False Cause I mean the fallacy usually referred to by the Latin phrase non causa pro causâ (the non-cause taken for the cause). In this fallacy we assume, without sufficient grounds, that one thing is the cause of another. People still blame a change in the weather on the new moon or full moon that came shortly before it, although it has been shown over and over again that the moon can have no such effect. In earlier centuries, any plague or other public disaster that followed the appearance of a comet or an eclipse was thought to be its result. The Latin phrase post hoc ergo propter hoc ("after this, therefore because of this") describes exactly what is wrong with conclusions like these. We no longer fear signs and omens, but we still commit the fallacy often enough. We do it when we assume that all the prosperity of England comes from the national character, forgetting that the country's plentiful coal and its position on the sea have contributed to our material wealth. It is, no doubt, just as fallacious to give national character no importance at all; or to argue that because England has misgoverned Ireland in past centuries, all the present evils of Ireland are due to that misgovernment.

Last comes the rather trivial Fallacy of Many Questions. It is committed by anyone who combines two or three questions into one in such a way that no true answer can be given. I cannot think of a better example than the crude joke of asking, "Have you stopped beating your mother?" Questions just as unfair are constantly asked by barristers examining witnesses in court, and no one can fairly be required to answer Yes or No to every question that may be put to him. As Aristotle says, "Several questions put as one should be at once decomposed into their several parts. Only a single question admits of a single answer: so that neither several predicates of one subject, nor one predicate of several subjects, but only one predicate of one subject, ought to be affirmed or denied in a single answer."

Read Professor De Morgan's excellent and amusing Chapter on Fallacies, Formal Logic, Ch. xiii.

Whately's remarks on Fallacies, in his Elements of Logic, Book III, are often very original and sharp.

Questions on Fallacies

1. Classify the fallacies.

2. Explain the following expressions:

A dicto secundum quid ad dictum simpliciter; ignoratio elenchi; argumentum ad hominem; argumentum ad populum; petitio principii; circulus in probando; non sequitur; post hoc ergo propter hoc.

3. What is arguing in a circle, and what is a question-begging epithet?

4. What different meanings can be given to the following sentence by changing which word is stressed?

"Newton's discovery of gravitation is not generally believed to have been at all anticipated by several philosophers in England and Holland."

5. Point out the ways in which each of the following sentences could be misunderstood.

(1) He went to London and then to Brighton by the express train.

(2) Did you make a long speech at the meeting?

(3) How much is five times seven and nine?

Miscellaneous Examples

(Continued from the questions to Lesson 18.)

The following examples are partly sound arguments and partly false ones. The reader is asked to deal with them as follows:

1. If the example is not in a simple and complete logical form, put it into whatever complete form seems most fitting.

2. Decide whether it is a valid argument or a fallacy.

3. Give the exact name of the argument, or of the fallacy, as the case may be.

4. If it is a categorical syllogism, reduce it to the first figure.

5. If it is a hypothetical syllogism, state it in categorical form.

21. Elementary substances alone are metals. Iron is a metal; therefore it is an elementary substance.

22. No Athenians could have been Helots (the serfs of Sparta); for all the Helots were slaves, and all Athenians were free men.

23. Aristotle must have been a man of extraordinary industry; for only such a man could have produced his works.

24. Nothing is better than wisdom; dry bread is better than nothing; therefore dry bread is better than wisdom.

25. Pitt (William Pitt the Younger, the Prime Minister) was not a great and useful minister; for although he would have been one if he had carried out Adam Smith's doctrines of Free Trade, he did not carry out those doctrines.

26. Only the virtuous are truly noble; some who are called noble are not virtuous; therefore some who are called noble are not truly noble.

27. Ireland is idle and therefore starves; she starves, and therefore rebels.

28. No designing person ought to be trusted; engravers are designers by profession; therefore they ought not to be trusted.

29. Logic as the schoolmen (the philosophers of the medieval universities) practiced it proved a fruitless study; therefore logic as it is practiced today must be a fruitless study too.

30. Is a stone a body? Yes. Then is not an animal a body? Yes. Are you an animal? I think so. So you are a stone, since you are a body. (Lucian.)

31. "If ye were Abraham's children, ye would do the works of Abraham." (John 8:39.)

32. "He that is of God heareth God's words: ye therefore hear them not, because ye are not of God." (John 8:47.)

33. Muhammad was a wise lawgiver; for he studied the character of his people.

34. Everyone desires virtue, because everyone desires happiness.

35. His weakness of character might have been inferred from his fondness for favorites; for all weak princes have this failing. (De Morgan.)

36. He is brave who conquers his passions; he who resists temptation conquers his passions; so he who resists temptation is brave.

37. Suicide is not always to be condemned; for it is only voluntary death, and many of the greatest heroes of the ancient world gladly embraced voluntary death.

38. Since all metals are elements, the rarest of all the metals must be the rarest of all the elements.

39. The express train alone does not stop at this station; and since the last train did not stop, it must have been the express train.

40. Peel's remission of taxes was beneficial; the taxes Peel remitted were indirect taxes; therefore the remission of indirect taxes is beneficial. (Sir Robert Peel, as Prime Minister in the 1840s, abolished or cut many duties on goods.)

41. Books are a source both of instruction and of amusement; a table of logarithms is a book; therefore it is a source both of instruction and of amusement.

42. All desires are not blameworthy; all desires are liable to excess; therefore some things liable to excess are not blameworthy.

43. Whoever intentionally kills another person should suffer death; so a soldier who kills his enemy should suffer death.

44. Schemers are unfit to be trusted; this man has formed a scheme; therefore he is unfit to be trusted.

45. Few towns in the United Kingdom have more than 300,000 inhabitants; and since all such towns ought to be represented by three members in Parliament, it is clear that few towns ought to have three representatives.

46. All the works of Shakespeare cannot be read in a day; therefore the play of Hamlet, being one of the works of Shakespeare, cannot be read in a day.

47. In moral matters we cannot stand still; therefore whoever does not go forward is sure to fall behind.

48. The people of the country are suffering from famine; and since you are one of the people of the country, you must be suffering from famine.

49. Substances that are lighter than water can float on it; the metals that can float on it are potassium, sodium, lithium, and so on; therefore potassium, sodium, lithium, and so on, are lighter than water.

50. The laws of nature must be discovered by Deduction, Traduction or Induction; but the first two are not enough for the purpose; therefore the laws of nature must be discovered by Induction.

51. A successful author must be either very hard-working or very talented; Gibbon was very hard-working; therefore he was not very talented.

52. You are not what I am; I am a man; therefore you are not a man.

53. The holder of some shares in a lottery is sure to win a prize; and since I am the holder of some shares in a lottery, I am sure to win a prize.

54. Gold and silver are wealth; and therefore a decrease of the gold and silver in the country by exporting it is a decrease of the wealth of the country.

55. Overly credulous people ought never to be believed; and since the Ancient Historians were in many instances overly credulous, they ought never to be believed.

56. Some mineral compounds are not decomposed by heat; all organic substances are decomposed by heat; therefore no organic substances are mineral compounds.

57. Whatever schools exclude religion are irreligious; non-sectarian schools do not allow the teaching of religious creeds; therefore they are irreligious.

58. Night must be the cause of day; for it always comes before it.

59. The ancient Greeks produced the greatest masterpieces of eloquence and philosophy; the Lacedaemonians (the Spartans) were ancient Greeks; therefore they produced the greatest masterpieces of eloquence and philosophy.

60. All presuming men are contemptible; this man, therefore, is contemptible; for he presumes to believe that his opinions are correct.

61. If a substance is solid it has elasticity, and so it does if it is liquid or gaseous; but all substances are either solid, liquid or gaseous; therefore all substances have elasticity.

62. If Parr's Life Pills (a popular patent medicine) are of any value, those who take them will improve in health; now my friend, who has been taking them, has improved in health; therefore they are of value.

63. He who calls you a man speaks truly; he who calls you a fool calls you a man; therefore he who calls you a fool speaks truly.

64. Whoever is most hungry eats most; whoever eats least is most hungry; therefore whoever eats least eats most.

65. Whatever produces drunkenness should be prohibited; the use of spirits (strong liquor) causes drunkenness; therefore the use of spirits should be prohibited.

66. What we eat grew in the fields; loaves of bread are what we eat; therefore loaves of bread grew in the fields.

67. If light consisted of material particles, it would have momentum; so it cannot consist of material particles, for it does not have momentum.

68. Everything that is morally right is allowed by law; indulgence in pleasures is allowed by law; therefore indulgence in pleasures is morally right.

69. All the trees in the park make a thick shade; this is one of them; therefore this tree makes a thick shade.

70. All visible bodies shine either by their own light or by reflected light. The moon does not shine by its own, therefore it shines by reflected light; but the sun shines by its own light, therefore it cannot shine by reflected light.

71. Honesty deserves reward, and a Black man is a fellow human being; therefore an honest Black man is a fellow human being who deserves reward.

72. Nearly all the satellites revolve around their planets from west to east; the moon is a satellite; therefore it revolves around its planet from west to east.

73. Italy is a Catholic country and is full of beggars; France is also a Catholic country, and therefore is full of beggars.

74. Every law is either useless or causes harm to someone; now a law that is useless ought to be abolished; and so ought every law that causes harm; therefore every law ought to be abolished.

75. The end of a thing is its perfection; death is the end of life; therefore death is the perfection of life.

76. When we hear that all righteous people are happy, it is hard not to exclaim: What! Are all the unhappy people we see to be thought unrighteous?

77. I am offered a sum of money to help this person get the office he wants; to help a person is to do him good, and no rule of morality forbids doing good; therefore no rule of morality forbids me to take the money for helping the person.

78. Ruminant animals (the cud-chewers) are those which have cloven hoofs, and they usually have horns; the extinct animal that left this footprint had a cloven hoof; therefore it was a ruminant animal and had horns. And again, since no beasts of prey are ruminant animals, it cannot have been a beast of prey.

79. We must either indulge our vicious tendencies or resist them; the first course will involve us in sin and misery; the second requires self-denial; therefore we must either fall into sin and misery or practice self-denial.

80. The stonemasons benefit from the masons' union; the bricklayers from the bricklayers' union; the hatmakers from the hatmakers' union; in short, every trade from its own union; so it is clear that if all workmen had unions, all workmen would benefit from them.

81. Every moral aim requires the rational means of reaching it; these means are the establishment of laws; and since happiness is the moral aim of human beings, it follows that reaching happiness requires the establishment of laws.

82. Whoever can swim need not despair of flying; for to swim is to fly in a coarser fluid, and to fly is to swim in a finer one.

83. If the Helvetii went through the country of the Sequani, they were sure to meet various difficulties; and if they went through the Roman province, they faced the danger of opposition from Caesar; but they had to go one way or the other; therefore either they were sure to meet various difficulties, or they faced the danger of opposition from Caesar. (Caesar, De Bello Gallico, Book I, 6.)

84. "Riches are for spending, and spending for honour and good actions; therefore extraordinary expense must be limited by the worth of the occasion." (Bacon.)

85. If light is not refracted near the surface of the moon, there cannot be any twilight there; but if the moon has no atmosphere, light is not refracted near its surface; therefore if the moon has no atmosphere, there cannot be any twilight.

86. The preservation of society requires exchange; whatever requires exchange requires a fair valuation of property; this requires the adoption of a common measure; so the preservation of society requires the adoption of a common measure.

87. The fact that the various species of brute animals were created to prey on one another proves that the human species was intended to prey on them.

88. The more correct the logic, the more certainly the conclusion will be wrong if the premises are false. Therefore, where the premises are wholly uncertain, the best logician is the least safe guide.

89. If our rulers could be trusted always to look to the best interests of their subjects, monarchy would be the best form of government; but they cannot be trusted; therefore monarchy is not the best form of government.

90. If people were prudent, they would act morally for their own good; if they were benevolent, they would do so for the good of others. But many people will not act morally, either for their own good or for that of others; such people, therefore, are not prudent or benevolent.

91. Whoever bears arms at the command of the magistrate does what is lawful for a Christian; the Swiss in the French service, and the British in the American service, bore arms at the command of the magistrate; therefore they did what was lawful for a Christian. (Whately.)

92. "A man that hath no virtue in himself ever envieth virtue in others; for men's minds will either feed upon their own good or upon others' evil; and who wanteth [lacks] the one will prey upon the other." (Bacon.)

93. The object of war is lasting peace; therefore soldiers are the best peacemakers.

94. Confidence in promises is essential to the dealings of human life; for without it most of our conduct would be left to chance. But there could be no confidence in promises if people were not obliged to keep them; therefore the obligation to keep promises is essential to the same ends, and in the same degree.

95. If the majority of those who use pubs (public houses) are prepared to close them, legislation is unnecessary; but if they are not prepared for such a measure, then to force it on them by outside pressure is both dangerous and unjust.

96. Whoever believes himself always to be in the right in his opinion lays claim to infallibility; you always believe yourself to be in the right in your opinion; therefore you lay claim to infallibility. (Whately.)

97. If we never find skins except as the coverings of animals, we may safely conclude that animals cannot exist without skins. If color cannot exist by itself, it follows that nothing colored can exist without color either. In the same way, if language without thought is unreal, thought without language must be unreal too.

98. No soldiers should be brought into the field who are not well qualified to do their part; none but veterans are well qualified to do their part; therefore none but veterans should be brought into the field. (Whately.)

99. The minimum visibile is the smallest size that can be seen; no part of it alone is visible, and yet every part of it must affect the mind in order for it to be visible; therefore every part of it must affect the mind without being visible.

100. The scarlet poppy belongs to the genus Papaver, of the natural order Papaveraceae; this order is part of the subclass Thalamiflorae, which belongs to the great class of Dicotyledons. So the scarlet poppy is one of the Dicotyledons.

101. Improbable events happen almost every day; but what happens almost every day is a very probable event; therefore improbable events are very probable events. (Whately.)

Lesson 22: The Quantification of the Predicate

In the last three lessons the syllogism has been explained almost exactly as it has been taught for more than two thousand years. Geometry has been taught in the way and the order first laid down by the ancient Greek writer Euclid, and in just the same way logic has been taught very nearly as Aristotle taught it, around the year 335 B.C.

But within the last few years, teachers in England have at last come to the conclusion that Euclid's ideas of geometry are not as perfect as they might be. And over the last thirty or forty years it has gradually become clear that Aristotle's syllogism is not an absolutely perfect system of logical deduction either. In fact, certain eminent writers, especially Sir William Hamilton, Professor De Morgan, Archbishop Thomson and George Boole, have shown that we need to make improvements from the very foundations of the science.

This reform in logic goes by the rather mysterious name of the quantification of the predicate. But a reader who has found no impossible difficulty in the lessons so far need not fear one here. To quantify the predicate simply means to state whether the whole of the predicate, or only part of it, agrees with or differs from the subject. Take this proposition:

	"All metals are elements."

Here the subject is quantified, but the predicate is not. We know that all metals are elements, but the proposition does not say plainly whether the metals make up the whole of the elements or not. Now take the quantified proposition

	"All metals are some elements."

The little word "some" says clearly that the metals really form only a part of the elements. Aristotle avoided using any mark of quantity on the predicate. As we have seen, he assumed that every affirmative proposition has a particular predicate, like the example just given, and that only negative propositions have a distributed, or universal, predicate. The fact is, however, that he was entirely wrong about this. As a result he shut out of his system an infinite number of affirmative propositions that are universal in both terms. It is true that "All equilateral triangles are all equiangular triangles," but this proposition could have appeared in his system only in the mutilated form

	"All equilateral triangles are equiangular."

A proposition such as

	"London is the capital of England,"

or "Iron is the cheapest metal," had no proper place at all in his syllogism. In each of these, both terms are singular and identical with each other, and so both are universal.

As soon as we allow the quantity of the predicate to be stated, the forms of reasoning become much simpler. Let us first look at the process of conversion. In our lesson on that subject we had to distinguish between conversion by limitation and simple conversion. Now a single process, simple conversion, is enough for every kind of proposition. So the quantified proposition of the form A,

	"All metals are some elements,"

is converted simply into

	"Some elements are all metals."

The particular affirmative proposition

	"Some metals are some brittle substances"

becomes, just by swapping the terms around,

	"Some brittle substances are some metals."

The particular negative proposition

	"Some men are not (any) trustworthy persons"

is also converted simply, into

	"Not any trustworthy persons are some men,"

though the result may seem less satisfactory in this form than in the affirmative form that follows:

	"Some men are some not-trustworthy persons,"

converted simply into

	"Some not-trustworthy persons are some men."

The universal negative proposition E is converted simply, as before. And finally we have a new affirmative proposition, universal in both subject and predicate, as in "All equilateral triangles are all equiangular triangles." This can obviously be converted simply into

	"All equiangular triangles are all equilateral triangles."

This doubly universal affirmative proposition is extremely common: every definition and every singular proposition is one. I may give as examples "Honesty is the best policy," "The greatest truths are the simplest truths," "Virtue alone is happiness below," and "Self-exaltation is the fool's paradise."

When affirmative propositions are put in quantified form, every immediate inference can easily be drawn from them by one rule: whatever we do to one term, we should do to the other. So from the doubly universal proposition "Honesty is the best policy" we infer that "what is not the best policy is not honesty," and also that "what is not honesty is not the best policy." In fact we can draw two contrapositives from this proposition. But the reader should carefully remember that from the ordinary, unquantified proposition A we can draw only one contrapositive (see Lesson 10). So if "metals are elements," we must not say that "what are not metals are not elements." But if we quantify the predicate, as in "All metals are some elements," we may infer that "what are not metals are not some elements." Immediate inference by added determinant and by complex conception can also be applied in either direction to quantified propositions, without any fear of the errors pointed out in Lesson 10.

Once we admit the mark of quantity in front of the predicate, it is clear that we double the number of propositions the syllogism must take in, because the predicate of each of the four propositions A, E, I, O may be either universal or particular. So we arrive at a list of eight conceivable kinds of proposition, set out in the following table.

The letters X and Y stand for any subject and any predicate, and by putting in various terms the reader can easily make up propositions of each kind. The symbolic letters on the left were proposed by Archbishop Thomson as a convenient way of referring to each of the eight propositions, and they are very well chosen. The doubly universal affirmative proposition is called U. The simple converse of A is called Y. The Greek letter η (Eta, ē) is given to the proposition we get by changing the universal predicate of E into a particular predicate, and the Greek ω (Omega, ō) to the proposition we get in the same way from O. Sir William Hamilton uses all eight of these propositions, but Archbishop Thomson considers that two of them, η and ω, are never really used. It is a remarkable fact that a complete table of these eight propositions was given by Mr. George Bentham in a work called Outline of a New System of Logic, published in 1827, several years before the earliest of Hamilton's writings on logic. But Mr. Bentham considered that some of the propositions can hardly be told apart from others: Y from A, of which it is the simple converse, for instance, or η from O.

Even using only the two extra propositions U and Y brought in by Thomson extends the list of possible syllogisms a great deal, to 62 in all, without counting the fourth figure, which Hamilton and Thomson do not use. When all eight propositions are admitted, we have to extend the list of possible syllogisms to 12 affirmative and 24 negative moods in each of the first three figures. All of these moods are conveniently set out in the table below, which Archbishop Thomson gives at p. 188 of his Laws of Thought.

	Table of Moods of the Syllogism.

Sir William Hamilton also invented a curious system of notation for showing all the moods of the syllogism clearly. He always used the letter M for the middle term of the syllogism, and the two letters C and Γ (the Greek capital letter Gamma) for the two terms that appear in the conclusion. He showed the copula of the proposition by a line thickened toward the subject, so that C ——— M means "C is M." To show the quantity of the terms, Hamilton put a colon (:) between the term and the copula when the quantity is universal, and a comma (,) when it is particular. In this way we can easily express the following affirmative propositions:

and so on. Any affirmative proposition can be turned into the corresponding negative proposition by drawing a stroke through the line that stands for the copula, as in the

Any syllogism can be shown by putting M, the middle term, in the center and joining it on each side to the other terms. The copula representing the conclusion can then be put underneath. Barbara is expressed like this:

The negative mood Celarent looks like this:

Cesare, in the second figure, is shown like this:

Sir William Hamilton also proposed a new law, or supreme canon, of the syllogism, by which the validity of every form of the syllogism could be tested. He stated it in these words: "What worse relation of subject and predicate subsists between either of two terms and a common third term, with which both are related, and one at least positively so—that relation subsists between these two terms themselves."

By a "worse" relation, Sir William means that a negative relation is worse than an affirmative one, and a particular relation worse than a universal one. So this canon expresses the rules that if there is a negative premise the conclusion must be negative, and if there is a particular premise the conclusion must be particular. Special canons were also worked out for each of the three figures. But in making the system so complicated, the advantages of the quantified form of proposition seem to be lost.

Professor De Morgan also discovered the advantages of the quantified predicate, and invented a system very different from Hamilton's. It is fully explained in his Formal Logic, in The Syllabus of a new System of Logic, and in various important papers on the syllogism in the Transactions of the Cambridge Philosophical Society. These works also give a complete explanation of the "Numerically Definite Syllogism." De Morgan pointed out that two particular premises can often give a valid conclusion, provided that the actual quantities of the two terms are stated and that, added together, they exceed the quantity of the middle term. Suppose a majority of a public meeting vote for the first resolution, and a majority also vote for the second. Then it follows necessarily that some who voted for the first also voted for the second. The two majorities added together are more than the whole number of people at the meeting, so they cannot be made up of entirely different people. They may, indeed, be made up of exactly the same people. But all we can deduce from the premises is that the excess of the two majorities added together over the number at the meeting must have voted for both resolutions. Hamilton said that this kind of inference depends on ultra-total distribution, and the name Plurative Propositions has been proposed for all those propositions that give a definite idea of what fraction or number of the subject the assertion is about.

T. Spencer Baynes, Essay on the new Analytic of Logical Forms; Edinburgh, 1850.

Professor Bowen's Treatise on Logic or the Laws of Pure Thought, Cambridge, U.S., 1866 (Trübner and Co.), gives a full and excellent account of Hamilton's logic.

Questions and Exercises

1. What does the quantification of the predicate mean?

2. Give each of the following propositions its proper symbol, and give the symbol of its converse:

(1) Knowledge is power.

(2) Some rectangles are all squares.

(3) Only the honest ultimately prosper.

(4) Princes have only their titles for their glories.

(5) In man there is nothing great but mind.

(6) The end of philosophy is the detection of unity.

3. Draw all the contrapositive propositions and immediate inferences you can from the following propositions:

(1) London is a great city.

(2) London is the capital of England.

(3) All ruminant animals are all cloven-footed animals.

(4) Some members of parliament are all the ministers.

4. Write out the moods Baroko, Darapti, Felapton and Bokardo in Hamilton's notation.

Lesson 23: Boole's System of Logic

In an elementary book like this one, it would be quite impossible to give any real idea of the system of indirect inference first discovered by the late George Boole, Professor of Mathematics at Queen's College, Cork. As I mentioned in the last lesson, his system was founded on the Quantification of the Predicate. But Boole regarded Logic as a branch of Mathematics, and he believed he could reach every possible inference by the principles of algebra. The process he actually used is very obscure and difficult, and so far hardly anyone has tried to bring it into elementary textbooks of Logic.

I have been able to reach exactly the same results as Boole without using any mathematics at all. The very simple process I am going to describe can hardly be called Boole's logic in the strict sense, but it is very similar to it, and it can prove everything Boole proved. This Method of Indirect Inference is founded on the three primary Laws of Thought stated in Lesson 14. Any reader who thought those laws were useless truisms may be surprised to find how wide and elegant a system of deduction can be built on them.

The law of excluded middle lets us assert that anything must either have a given quality or not have it. If the thing is iron, say, and the quality is combustibility, anyone can see that

	"Iron is either combustible or incombustible."

We can repeat this division into alternatives as often as we like. Let Book be the class of things to be divided, and let English and Scientific be two qualities. Then any book must be either English or not English. An English book, in turn, must be either Scientific or not Scientific, and the same goes for books that are not English. So we can divide books straight away into four classes:

	Books, English and Scientific.
	Books, English and not-Scientific.
	Books, not-English and Scientific.
	Books, not-English and not-Scientific.

We may call this an exhaustive division of the class Books. There is no possible book that does not fall into one or other of these four divisions, for the simple reason that if it does not fall into any of the first three, it must fall into the last. The process can go on without end, as long as anyone can suggest a new circumstance to divide by. We might divide each class again according to whether the books are octavo (a standard mid-sized format) or not octavo, bound or unbound, published in London or elsewhere, and so on. This process of twofold division is really the process of Dichotomy described in Lesson 12. We shall call it the development of a term, because it always lets us develop the greatest number of alternatives that need to be considered.

As a general rule, it is unlikely that all the alternatives unfolded, or developed, in this way can actually exist. The next step is to find out how many do or may exist. The Law of Contradiction asserts that nothing can combine contradictory attributes or qualities, so if we meet any term that contradicts itself in this way, we are entitled to strike it out of the list at once. Now take our old example of a syllogism:

	Iron is a metal;
	All metals are elements;
	Therefore iron is an element.

We can easily prove this conclusion by the indirect method. If we develop the term iron, we get four alternatives:

	Iron, metal, element.
	Iron, metal, not-element.
	Iron, not-metal, element.
	Iron, not-metal, not-element.

Now compare each of these alternatives with the premises of the syllogism, and it becomes clear that several of them cannot exist. We are told that iron is a metal, so no class of things "iron, not-metal" can exist. The first premise therefore lets us strike out both of the last two alternatives, since they combine iron with not-metal. The second alternative combines metal with not-element. But the second premise tells us that "all metals are elements," so that one must be struck out too. Only one alternative remains that can exist if the premises are true. And since there cannot possibly be any alternatives beyond the ones we have considered, it follows by strict demonstration that iron occurs only together with the qualities of metal and element; in short, that iron is an element.

But we can prove more than the ordinary syllogistic conclusion. We can prove any other conclusion that can be drawn from the same premises. In fact, the syllogistic conclusion is usually only one of many that can be got from a given set of premises. Suppose, for instance, that we want to know what the term or class not-element is like, as far as the premises we have just used can tell us. We can develop the alternatives of this term just as we did those of iron, and we get the following:

	Not-element, iron, metal.
	Not-element, iron, not-metal.
	Not-element, not-iron, metal.
	Not-element, not-iron, not-metal.

Compare these combinations with the premises, as before. The first plainly cannot exist, because all metals are elements, and the third cannot exist for the same reason. The second is excluded as well, because iron is a metal and cannot exist together with the qualities of not-metal. So only one combination remains to represent the class we wanted, namely

	Not-element, not-iron, not-metal.

So the premises teach us that every not-element is not a metal and is not iron.

For another example of this kind of deduction, I will take a case of the Disjunctive Syllogism in the negative mood:

	A fungus is either plant or animal,
	A fungus is not an animal;
	Therefore it is a plant.

If we develop all the possible ways in which fungus, plant and animal can be combined, we get, for the term fungus:

	(1) Fungus, plant, animal.
	(2) Fungus, plant, not-animal.
	(3) Fungus, not-plant, animal.
	(4) Fungus, not-plant, not-animal.

Of these, the 4th cannot exist, because according to the premise a fungus must be a plant, or, if not a plant, an animal. The 1st and 3rd cannot exist either, because the minor premise tells us that a fungus is not an animal. That leaves only the second combination,

	Fungus, plant, not-animal,

and from it we learn the syllogistic conclusion that "a fungus is a plant."

The chief strength of this way of deducing is that it is not tied to a fixed series of forms, as the syllogism is. Without any extra rules, it applies to every kind of proposition or problem that can be conceived and stated. There may be any number of premises, and they may contain any number of terms. To get any possible inference, all we have to do is develop the term we want into all its alternatives and then see how many of them agree with the premises. Whatever is left after this examination is, necessarily, the description of the term. The method's only drawback is that the number of alternatives to examine grows very fast as the number of terms grows, and writing them all out soon becomes tedious. We can shorten the work by putting single letters in place of the terms, somewhat as in algebra. We may take A, B, C, D, and so on, to stand for the affirmative terms, and a, b, c, d, and so on, for the corresponding negative ones. For a first example, take the premises:

	Organic substance is either vegetable or animal.
	Vegetable substance consists mainly of carbon, hydrogen, and nitrogen.
	Animal substance consists mainly of carbon, hydrogen, and nitrogen.

It would take a long time to write out all the combinations of the four terms that occur here. But if we put letters in place of them, like this,

	A = organic substance,
	B = vegetable substance,
	C = animal substance,
	D = consisting mainly of carbon, hydrogen, and nitrogen,

we can easily set out all the combinations that can belong to the term A:

Now the premises amount to the statements that

	A must be either B or C,
	B must be D,
	C must be D.

Combinations (7) and (8) are inconsistent with the first premise; combinations (2) and (4) with the second premise; and (6) is inconsistent with the third premise. That leaves only

	ABCD
	ABcD
	AbCD.

From this we learn at once that "organic substance (A) always consists mainly of carbon, hydrogen and nitrogen," because it always occurs together with D. The reader may notice that the combination ABCD implies that an organic substance may be both vegetable (B) and animal (C). If the first premise is taken to mean that this is impossible, then of course this combination should be struck out as well. Whether the alternatives of a disjunctive proposition can exist together or not is an unsettled question (see Lesson 19), but I much prefer the view that they can. And as a matter of fact, it is quite likely that there are very simple kinds of living things that cannot be said for certain to be only vegetable or only animal, but share the nature of both.

To show the power of this system on a harder problem, let us take the premises Boole treated in his Laws of Thought, p. 125:

"Similar figures consist of all whose corresponding angles are equal, and whose corresponding sides are proportional.

Triangles whose corresponding angles are equal have their corresponding sides proportional; and vice versa.

Triangles whose corresponding sides are proportional have their corresponding angles equal."

Now suppose we choose our letters as follows:

	A = similar figure,
	B = triangle,
	C = having corresponding angles equal,
	D = having corresponding sides proportional.

Then the premises turn out to amount to the statements that

	A is identical with CD,

and that

	BC is identical with BD.

In other words, all A's must be CD's, and CD's must be A's; all BC's must be BD's, and all BD's must be BC's.

The letters can be combined in 16 possible ways, shown in the following table:

Comparing each of these combinations with the premises, we see that ABCd, ABcD, ABcd and others must be struck out, because every A has to be CD as well. The combinations aBCD and abCD are struck out because every CD must also be A. Again, aBCd is inconsistent with the condition that every BC must also be BD. If the reader carefully carries the same examination through to the end, only six combinations will be left that agree with all the premises:

From these combinations we can draw any description we like of the classes of things that agree with the premises. The class A, or similar figures, is represented by only two combinations, or alternatives. The negative class a, or dissimilar figures, is represented by four. So we may draw the following conclusion: "Dissimilar figures consist of all triangles that do not have their corresponding angles equal and their sides proportional (aBcd), and of all figures that are not triangles and that have either their angles equal but their sides not proportional (abCd), or their corresponding sides proportional but their angles not equal (abcD), or neither their corresponding angles equal nor their corresponding sides proportional (abcd)."

Anyone who works this method of inference soon sees that it runs in a very simple, mechanical way, and that its only drawback is the large number of alternatives, or combinations, to be examined. So I have invented several ways of cutting down the labor. The simplest is to engrave the series of 16 combinations in the table above, which turn up again and again in problems, along with larger and smaller sets, on an ordinary writing slate. Then the excluded combinations can easily be struck out with an ordinary slate pencil, and the slate can still be used again for any later logical question. A second device, which I have called the "Logical abacus," is made by printing the letters on slips of wood fitted with pins, arranged so that any part or class of the combinations can be picked out mechanically with very little trouble. A logical problem is then solved by the hand rather than by the head. More recently, though, I have reduced the system to a completely mechanical form, and so built the whole indirect process of inference into what may be called a Logical Machine. On the front of the machine are some movable wooden rods carrying the set of 16 combinations of letters shown in the table above. At the foot are 21 keys like those of a piano. Eight keys toward the left are marked with the letters A, a, B, b, C, c, D, d, and they stand for these terms when they occur in the subject of a proposition. Eight more keys toward the right stand for the same letters, or terms, when they occur in the predicate. The copula of a proposition is represented by a key in the middle of the row, and the full stop by a key at the far right. Two other keys serve for the disjunctive conjunction or, one for when it occurs in the subject and one for when it occurs in the predicate. Now let the letters stand for the terms of a syllogism, or of any other logical argument, and press the keys of the instrument in exactly the order of the words of the premises. The 16 combinations will be sorted and arranged by this so that at the end only the possible combinations remain in view. Any question can then be put to the machine, and the combinations that remain will give an infallible answer. So the internal workings of the machine actually perform the work of inference that, in Boole's system, was done by a very complicated mathematical calculation. I should add that there is one last key at the far left, which wipes out all the previous operations and puts all the combinations back in their original places, so that the machine is ready for a new problem.

An account of this logical machine can be found in the Proceedings of the Royal Society for January 20, 1870; on that day the machine was shown working to the Fellows of the Society. The principles of the method of inference described here are set out more fully in The Substitution of Similars and in the Principles of Science, which I published in 1869 and 1874. I may add that each of these works contains certain views about the real nature of the process of inference that I do not think it wise to bring into an elementary book like this one, because they are speculative. The process of inference I have taken from Boole's system, on the other hand, is so self-evident, and so clearly proved true by being reduced to a mechanical form, that I do not hesitate to bring it to the reader's attention.

Footnote: The Substitution of Similars, the true Principle of Reasoning, derived from a modification of Aristotle's Dictum. Macmillan and Co. 1869.

Footnote: The Principles of Science: a Treatise on Logic and Scientific Method. 2 vols. Macmillan and Co.

George Boole, Mathematical Analysis of Logic, 1847.

An Investigation of the Laws of Thought. London, Walton and Maberly, 1854.

Questions and Exercises

1. Apply this system of inference to prove the syllogisms in Cesare and Camestres given in Lesson 16.

2. Show that if no A's are B's, then no B's are A's; and that if all A's are all B's, then all not-A's are all not-B's.

3. Develop the term substance with respect to the terms vegetable, animal, organic. Then pick out the combinations that agree with these premises: "What is vegetable is not animal but is organic; what is animal is organic."

4. Test whether this argument is valid: "Good always triumphs, and vice always fails; therefore the victor cannot be wrong, nor the vanquished right."

5. It is known of a certain class of things that:

(1) Where the quality A is, B is not.

(2) Where B is, and only where B is, C and D are. What can we infer from these premises about the class of things in which A is not present but C is present?

6. If all A's are B's, all B's are C's, and all C's are D's, show that all A's are D's, and that all not-D's are not A's.

Lesson 24: On Method, Analysis and Synthesis

Many writers on logic have held that logical doctrine has a fourth part, in addition to the three parts that deal in turn with Terms, Propositions and Syllogisms. That fourth part is method. The doctrine of Judgment considers how terms are arranged and combined into propositions. The doctrine of the Syllogism considers how propositions are arranged so that they form arguments. In the same way, they said, there should be a fourth part, called Method, to govern how syllogisms are arranged and combined into a complete discourse. So method is defined as arranging the parts of a discourse so that the whole is as easy as possible to understand.

The famous Peter Ramus, who was killed in the Massacre of St. Bartholomew's Day (Paris, 1572), was the first to propose making method a part of the science of logic in this way. But we may well doubt whether any definite set of rules or principles can be given to guide us in arranging arguments. Every discourse has to be made of arguments arranged to suit the particular nature of its subject. Things vary without end in the way they need to be treated, and no general rules can be given for treating them. So the supposed general rules of method are no better than truisms: they tell us nothing we must not be supposed to know already. For example, we are told that in composing any discourse we should take care that:

1. Nothing should be missing or superfluous.

2. The separate parts should agree with each other.

3. Nothing should be treated unless it suits the subject or purpose.

4. The separate parts should be connected by suitable transitions.

But obviously the whole difficulty lies in deciding what is missing or superfluous, what is suitable or consistent. Rules of this kind simply tell us to do what we ought to do, without saying what that is.

All the same, there are certain general ways of treating any subject that can be clearly told apart, and the student of logic should understand them well. Logic cannot teach him exactly how and when to use each kind of method. But it can teach him what the methods are and what they can do, so that he is more likely to use them rightly. We must distinguish:

	1. The method of discovery,
	2. The method of instruction.

The method of discovery is the one we use in acquiring knowledge. It really consists of the processes of inference and induction by which we find out general truths from collecting and examining particular facts. This method will be the subject of most of our remaining lessons. The second method applies only when knowledge has already been acquired and expressed in the form of general laws, rules, principles or truths. Then all we have to do is get to know these, and learn the proper way to apply them to particular cases, and we will have a complete knowledge of the subject.

Take a student learning Latin, Greek, French, German, or any other well-known language. He is given a complete grammar and syntax that set out all the principles, rules and nature of the language. He receives these instructions and takes them as true on the authority of his teacher, or of the writer of the book. Once he has made them familiar, he has nothing left to do but combine and apply the rules in reading or writing the language. In short, he follows the method of instruction. But the scholar who has been handed some writings in an unknown language, and is trying to work out its alphabet, words, grammar and syntax, has to go about it in a completely different, opposite way. He doesn't have the laws of the grammar. He has words and sentences that obey those laws, and he has to detect the laws, if he can, by watching their effects on the written language. In short, he follows the method of discovery. It consists of a tedious comparison of letters, words and phrases, carried out so as to reveal the forms and combinations in which they most often occur. The process would be a strictly inductive one, of the kind I will partly illustrate in the lessons on induction. But it is far harder than the method of instruction, and it depends a great deal on making good use of conjecture and hypothesis, which takes a certain skill and inventiveness.

Exactly the same is true of investigating natural things and events. The principles of mechanics, such as the lever, the inclined plane and the other Mechanical Powers, or the Laws of Motion, seem fairly simple and obvious when they are explained to us in textbooks. But the early philosophers had no such books. They had only the Book of Nature, which sets out not the laws but the results of the laws. It took the most patient and skillful investigation, and hundreds of mistakes, before those laws were found out. Today it is very easy to understand the Copernican system of astronomy, which has the planets revolving around the sun in orbits of different sizes. Once we know the theory, we can easily see why the planets move in such different ways and appear in such different positions, and why they sometimes seem to stand still. It is easy to see, too, why on top of their own proper motions they all seem to go around the earth every day, because the earth turns on its axis daily. But all these changes were extremely puzzling to the ancients, who believed the earth stood still.

So the method of discovery begins with facts that are apparent to the senses, and it has the hard task of detecting the universal laws or general principles that only the intellect can grasp. It has been well said that the method of discovery goes from the things that are better known to us, that is, to our senses (nobis notiora), to the things that are simpler, or better known in nature (notiora naturæ). The method of instruction goes in the opposite direction. It begins with the things notiora naturæ and goes on to show or explain the things nobis notiora. The difference is almost like the difference between hiding and seeking. Whoever hid a thing knows where to find it. A discoverer is not in that position. He has no clue except what he may come across in his own careful and shrewd search.

Closely matching the distinction between the methods of discovery and instruction is the distinction between the methods of analysis and synthesis. It is very important that the reader understand clearly what these terms mean in their various uses. Analysis is the process of separating a whole into its parts, and synthesis is the combining of parts into a whole. An analytical chemist who is given a piece of mineral to examine may be able to separate completely the chemical elements it is made of, and find out what they are and in what proportions they are present. This is chemical analysis. At other times the chemist mixes carefully weighed amounts of certain simple substances and combines them into a new compound substance. This is chemical synthesis. Logical analysis and synthesis must not be confused with these physical actions. Even so, they are actions of the mind of a similar kind.

In logical synthesis we begin with the simplest possible notions or ideas and combine them. The best possible example is elementary geometry. In Euclid we begin with certain simple notions: points, straight lines, angles, right angles, circles and so on. Put three straight lines together and we make a triangle. Add to this the notion of a right angle, and we form the notion of a right-angled triangle. Join four equal lines at right angles to each other and we get the idea of a square. Now suppose we imagine such a square built on each side of a right-angled triangle, and reason from the necessary qualities of these figures. We discover that the two squares on the sides that contain the right angle must together be exactly equal to the square on the third side, as is shown in the 47th Proposition of Euclid's first book. This is a perfect example of combining simple ideas into more complex ones.

Often, though, in geometry we have to take the opposite course, analysis. We may be given a complicated geometrical figure, and in order to prove the properties it has, we may have to break it up into its separate parts and consider the properties of each part separately from the others.

A similar distinction between the analytic and the synthetic methods can be traced all through the natural sciences. By keeping exact records of how the weather looks and changes, we can easily build up an immense collection of facts. Each recorded fact involves a crowd of different circumstances occurring together. In any storm or shower of rain, for instance, we have to consider the direction and strength of the wind; the temperature and dampness of the air; the height and shapes of the clouds; the amount of rain that falls, or the lightning and thunder that come with it. If we tried to explain the changes of the weather by analysis alone, we would have to try breaking each storm or change of weather down into its separate circumstances, and comparing each with every other, to find out which circumstances usually go together. No doubt we could find out in this way, with considerable certainty, what kinds of clouds and what changes of wind, temperature, moisture and so on usually come before each kind of storm. In time we might even give some imperfect explanation of what goes on in the atmosphere.

But we could also make good use of the synthetic method. Earlier chemical investigations tell us that the atmosphere consists mainly of two permanent gases, oxygen and nitrogen, together with water vapor, whose amount varies a great deal. We can test by experiment what happens when portions of such air, at various degrees of dampness, are compressed or allowed to expand, or are mixed together, as often happens in the atmosphere. In this way we discover that whenever moist air is allowed to expand, cloud is produced, and perhaps drops of rain. Dr. Hutton, too, found that whenever cold moist air is mixed with warm moist air, cloud is again produced. We can safely argue from small experiments like these to what happens in the atmosphere. By putting together synthetically everything that the sciences of chemistry, mechanics and electricity tell us about air, wind, cloud and lightning, we can explain what happens in a thunderstorm far more completely than we could by simply watching what happens in the storm. Here, however, we are getting ahead of ourselves, into the methods of inductive investigation, which we must consider in the following lessons. It will turn out that induction is equivalent to analysis, and that the deductive kinds of reasoning we have dealt with in the earlier lessons are synthetic in character.

It has been said that the synthetic method usually corresponds to the method of instruction, and the analytic method to the method of discovery. But it is possible to discover new truths by synthesis and to teach old ones by analysis. Sir John Herschel, in his well-known Outlines of Astronomy, partly adopts the analytic method. He imagines a spectator who first surveys the appearances of the heavenly bodies and of the earth's surface and looks for an explanation. Then he leads him through a series of arguments showing that these appearances really point to the roundness of the earth, its turning on its own axis and its revolution around the sun, and its humble position as one of the smaller planets of the solar system. Mr. Norman Lockyer's Elementary Lessons in Astronomy is a clear example of the synthetic method of instruction. He starts by describing the sun, the center of the system, and then adds the planets and the other members of the system one after another, until at last we have the complete picture. The reader, who has for the time being taken everything on the writer's authority, then sees that the description matches the truth. Each method, we must admit, has its own advantages.

We must take careful note that the meaning of analysis, and so the meaning of synthesis, changes depending on whether we are looking at terms in intension or in extension. To divide, or analyze, a class of things in extension, I must add a quality, or difference. For example, I divide the class organism when I add the quality vegetable, and separate vegetable organism from what is not vegetable. So analysis in extension is the same process as synthesis in intension. And the other way around: whenever I separate, or analyze, a group of qualities, each part belongs to a larger class of things in extension. When I analyze the notion vegetable organism, and consider the notion organism apart from vegetable, I am obviously adding the whole class of animal organisms to the class I am considering. So analysis in intension is synthesis in extension. A reader who has thought carefully about Lessons 5 and 12 will probably see that this connection between the two processes is just a restatement of the law (see Lesson 5) that "as the intension of a term is increased, the extension is decreased."

To express the difference between knowledge gained deductively and knowledge gained inductively, the Latin phrases a priori and a posteriori are often used. By a priori reasoning we mean argument based on truths already known. A posteriori reasoning, on the other hand, works from the consequences of a general truth to infer what that general truth is. Many philosophers believe that the mind naturally possesses certain laws or truths that it must recognize in every act of thought. All such truths, if there are any, would be a priori truths. No one can doubt, for example, that in thinking we must always recognize the three Primary Laws of Thought discussed in Lesson 14. There we have a priori knowledge that "matter cannot both have weight and be without weight," or that "everything must be either self-luminous or not self-luminous." But no law of thought can force us to think that matter has weight while the luminous ether (the medium that was then believed to carry light) has no weight, or that Jupiter and Venus are not self-luminous but comets are to some extent self-luminous. No doubt these facts are necessary consequences of the laws of nature and the general make-up of the world. But since we are not born knowing all the secrets of creation, we have to learn them by observation, that is, by the a posteriori method.

Nowadays, however, the name a priori is not usually restricted to truths obtained entirely without the help of observation. Knowledge may start out a posteriori, and yet, once it has been held for a long time and has become as certain as anything can be, it may serve as the ground of deductions. It may then be said to give a priori knowledge. For example, all scientists now believe that force (in today's terms, energy) cannot be created or destroyed by any of the processes of nature. If this is true, the force that disappears when a bullet hits a target must be turned into something else, and on a priori grounds we can say that the result will be heat. It's true that we could easily learn the same thing a posteriori, by picking up pieces of a bullet that has just hit a target and noticing that they are warm. But a priori knowledge has a great advantage: we can often apply it where experiment or observation would be difficult. If I lift a stone and drop it, the most delicate instruments could hardly show that the stone was heated by hitting the ground. Yet on a priori grounds I know that it must have been, and I can easily calculate how much heat was produced. In the same way we know, without the trouble of observing it, that Niagara Falls and every other waterfall produce heat. This is a fair example of a priori knowledge, because no one, as far as I have heard, has tested the fact or proved it a posteriori. All the same, the knowledge is founded in the first place on the experiments of Mr. Joule, who measured, in certain well-chosen cases, how much force is equivalent to a given amount of heat. The reader should take care, though, not to confuse the meaning of a priori explained here with the meaning given to the words by the philosophers who hold that the mind possesses knowledge independently of all observation.

It is not hard to see that the a priori method is equivalent to the synthetic method (see earlier in this lesson) considered in intension, and the a posteriori method, of course, is equivalent to the analytic method. But the same difference is really expressed by the words deductive and inductive, and we will often need to consider it in the following lessons.

For general remarks on method, see the Port Royal Logic, Part IV.

Questions and Exercises

1. What place did earlier writers on logic give to method, and what are the rules of method?

2. Explain the expressions nobis notiora and notiora naturæ.

3. What kind of method is the usual method of instruction?

4. Prove that analysis in extension is synthesis in intension, using some of the series of terms in question 6 of Lesson 5 as illustrations.

5. Explain the exact meanings of the expressions a priori knowledge and a posteriori knowledge.

6. To which kind does our knowledge of the following facts belong?

(1) The light of the stars takes a long time to reach us.

(2) Vaccination protects against smallpox.

(3) A meteor becomes heated as it passes through the air.

(4) There must be either some inhabitants or no inhabitants on Jupiter.

Lesson 25: Perfect Induction and the Inductive Syllogism

In the earlier lessons we have been looking at deductive reasoning. Deduction combines two or more propositions synthetically and arrives at a conclusion that is less general than the premises: a proposition or truth that applies to fewer individual instances than the separate premises it was inferred from. Take the general truth that "metals are good conductors of heat," and combine it with the truth that "aluminum is a metal." A syllogism in the mood Barbara then lets me infer that "aluminum is a good conductor of heat." This proposition is about one metal only, so it is plainly less general than the premise, which was about every metal there is. Induction goes the other way. In induction we move from less general facts, or even from individual facts, to more general propositions or truths, or, as we will often call them, Laws of Nature. Once we know that Mercury moves around the sun in an elliptical orbit, and that Venus, the Earth, Mars, Jupiter and the rest do too, we can arrive at the simple, general truth that "all the planets move in elliptical orbits around the sun." That is an example of inductive reasoning.

It is true that we can also reason without making our conclusion either more or less general than the premises. For example:

	Snowdon is the highest mountain in England or Wales.
	Snowdon is not so high as Ben Nevis.
	Therefore the highest mountain in England or Wales is not so high as Ben Nevis.

Or again:

	Lithium is the lightest metal known.
	Lithium is the metal indicated by one bright red line in the spectrum.
	Therefore the lightest metal known is the metal indicated by a spectrum of one bright red line.

Footnote: Roscoe's Lessons in Elementary Chemistry.

In these examples every proposition is a singular proposition. Each one simply asserts that two singular terms are identical, so the generality never changes. Each conclusion applies to exactly the same sort of object as each premise does. This kind of reasoning has been given the apt name of traduction.

Induction is a much harder and much more important kind of reasoning than traduction, or even deduction. Its work is to discover the general laws or uniformities, the relations of cause and effect, in short all the general truths we can assert about the countless and very varied events that happen in the natural world around us. Most of our knowledge, and perhaps all of it, as some philosophers think, comes in the end from inductive reasoning. The mind, people plausibly say, does not come supplied with knowledge already made up into general propositions and stamped on it. Instead it is given powers of observation, comparison and reasoning. When these powers are well trained and exercised, they are enough to get us knowledge of the world outside us and of the world inside the human mind. Even when we argue synthetically and deductively from simple ideas and truths that seem to be ready-made in the mind, as we do in geometry, we may in fact have gathered those simple ideas and truths earlier, by observation or by a kind of induction we were hardly aware of. This is a disputed point, and I will not take a firm position on it here. But if it is true, then induction is the way all the materials of knowledge are brought into the mind and analyzed. Deduction will then be the almost equally important process by which we put that knowledge to use, and by which, as we will see, new and more complicated inductions become possible.

An induction, that is, one act of inductive reasoning, is called perfect when all the possible cases or instances the conclusion can refer to have been examined and listed in the premises. Usually it is impossible to examine every case, since some may occur in the future, or in distant parts of the earth, or in other regions of the universe. Then the induction is called imperfect. The statement that every month of the year is shorter than thirty-two days comes from perfect induction, and it is a certain conclusion. The calendar is a human institution, so we know beyond doubt how many months there are, and we can easily check that each of them is shorter than thirty-two days. But the statement that all the planets move around the sun in one direction, from west to east, comes from imperfect induction. There may be planets farther out than Neptune, the most distant planet known in the 1870s, and the statement would of course apply to such a planet too.

So there is obviously a great difference between perfect and imperfect induction. Imperfect induction includes some process that lets us make statements about things we have never seen or examined, or even known to exist. But we must also be careful to remember that no imperfect induction can give a certain conclusion. It may be highly probable, or nearly certain, that the unexamined cases will be like the ones we have examined, but it can never be certain. It is quite possible, for instance, that a newly found planet might go around the sun in the opposite direction from the others. Among the satellites of the planets, more than one exception of this kind has already been discovered, and science has made mistakes again and again by expecting every new case to be exactly like the old ones. Imperfect induction, then, gives only a certain degree of probability, or likelihood, that all the instances will agree with the ones examined. Perfect induction, on the other hand, gives a necessary and certain conclusion, but it asserts nothing beyond what the premises already asserted.

Mr. J. S. Mill, in fact, disagrees with almost every other logician here. He holds that perfect induction is wrongly called induction at all, because it does not lead to any new knowledge. He defines induction as inference from the known to the unknown, and he counts the unexamined cases that seem to be brought into our knowledge as the only thing the reasoning gains. So perfect induction seems to him to have no scientific value whatever, because its conclusion merely says the premises over again in a shorter form, a mere summing up. I would point out, though, that even if perfect induction were nothing more than a way of abbreviating, it would still be very important, and we need it constantly in science and in everyday life. Without it we could never make a comprehensive statement at all. We would have to list every particular case. Suppose we examined the books in a library and found that they were all English books. Without perfect induction we could not sum up our results in the single proposition "all the books in this library are English books." We would have to go through the list of books every time we wanted to tell anyone what the library held. The truth is that the power to express a great number of particular facts in a very small space is essential to the progress of science. The whole art of arithmetic is nothing but a set of processes for shortening addition and subtraction, so that we can handle a great number of units in a very short time. In just the same way, perfect induction is absolutely necessary if we are to handle a great number of particular facts in a very small space.

Perfect induction is usually set out in the form of an inductive syllogism, as in this example:

	Mercury, Venus, the Earth, and so on, all move around the sun from west to east.
	Mercury, Venus, the Earth, and so on, are all the known planets.
	Therefore all the known planets move around the sun from west to east.

This argument is a true perfect induction. The conclusion makes a statement only about all the known planets, which shuts out any reference to planets that might be discovered later, and we may suppose that all the known planets have been listed in the premises. The argument looks like a syllogism in the third figure, namely Darapti, with the group of known planets as the middle term. In reality, though, it is not an ordinary syllogism. The minor premise does not say that Mercury, Venus, the Earth, Neptune and so on are contained among the known planets. It says that they are those planets, that they are identical with them. So this premise is a doubly universal proposition, of a kind (see Lesson 22) that the Aristotelian syllogism does not recognize. That is why the conclusion here is a universal proposition, something not allowed in the third figure of the syllogism.

As another example of perfect induction, we can take this:

	January, February, ... December, each contain less than 32 days.
	January ... December are all the months of the year.
	Therefore all the months of the year contain less than 32 days.

Sir William Hamilton rejected the idea completely, but it seems worth asking whether the inductive syllogism is not really a disjunctive syllogism. I would be inclined to set out the last example in this form:

A month of the year is either January, or February, or March, ... or December. But January has less than 32 days; and February has less than 32 days; and so on until we come to December, which has less than 32 days.

It clearly follows that a month, whichever it is, must have less than 32 days. There are only twelve possible cases, and in each one this is asserted. The fact is that the major premise of the syllogism above is a compound sentence with twelve subjects, so it is equivalent to twelve separate logical propositions. The minor premise is either a disjunctive proposition, as I have set it out, or something quite different from anything we have met elsewhere.

From perfect induction we will have to go on to imperfect induction. But logicians do not agree about what entitles us to take only some of the instances and conclude that what is true of those is true of all. Suppose we take the example found in many books and say:

	This, that, and the other magnet attract iron;
	This, that, and the other magnet are all magnets;
	Therefore all magnets attract iron.

Here we are obviously using a false minor premise, because this, that and the other magnet that we examined cannot possibly be all the magnets in existence. However we put it, there has to be an assumption that the magnets we examined are a fair sample of all magnets, so that what we find in some we may expect in all. Archbishop Whately thinks this assumption ought to be stated in one of the premises, and he sets out induction as a syllogism in Barbara, like this:

	That which belongs to this, that, and the other magnet, belongs to all;
	Attracting iron belongs to this, that, and the other;
	Therefore it belongs to all.

This is no doubt a correct statement of the assumption made in an imperfect induction. But it does not explain in the least what entitles us to make the assumption, or in what circumstances such an assumption is likely to turn out true. Some writers have claimed that there is a principle called the Uniformity of Nature, which lets us affirm that what has often been found true of anything will go on being found true of the same sort of thing. We must notice, though, that if there is such a principle, it has exceptions. Many facts that held true up to a certain point have later turned out not to be always true. For example, there was a wide and unbroken induction that seemed to show that all the satellites in the planetary system went around their planets in one and the same direction. Yet when the satellites of Uranus were discovered, they were found to move in a retrograde direction, that is, in the opposite direction from every satellite known before. The satellite of Neptune, discovered more recently (the only one known in the 1870s), has the same peculiarity.

We can leave to the next lesson the question of how the degree of certainty that induction gives varies from one branch of knowledge to another.

The advanced student will find it useful to consult the following: Mansel's Aldrich, Appendix, Notes G and H. Hamilton's Lectures on Logic, Lecture XVII, and Appendix VII, On Induction and Example, Vol. II, p. 358. J. S. Mill's System of Logic, Book III, Chap. 2, Of Inductions improperly so-called.

Questions and Exercises

1. Define deduction, induction and traduction, and distinguish them from one another.

2. Find an example of reasoning by traduction.

3. Distinguish perfect and imperfect induction.

4. How does Mr. Mill define induction, and what does he think of perfect induction?

5. What is the use of perfect induction?

6. Construct some examples of the inductive syllogism, and show that they can be put into a disjunctive form.

Lesson 26: Geometrical and Mathematical Induction, Analogy and Example

We now have to look very carefully at what Imperfect Induction rests on. Perfect Induction gives no trouble, because every case that can fall under the general conclusion is listed in the premises. In fact the conclusion holds no information that the premises did not already give. In this respect the Inductive Syllogism fits perfectly with the general principles of deductive reasoning. Those principles require that the information in the conclusion be drawn only from the data: we merely unfold what the premises contain implicitly, and turn it into an explicit statement.

In Imperfect Induction the process seems to be of a completely different kind, because the cases we gain knowledge about may be infinitely more numerous than the cases we gain it from. Let us first consider geometrical reasoning, which closely resembles inductive reasoning. In the fifth proposition of the first book of Euclid we prove that the angles at the base of an isosceles triangle are equal to each other. We do it by taking one particular triangle as an example. We are given a figure, asked to regard it as having two equal sides, and shown conclusively that if the sides really are equal, the angles opposite them must be equal too. But Euclid says nothing about any other isosceles triangle. He treats one single triangle as a sufficient specimen of all isosceles triangles, and we are asked to believe that whatever is true of it is true of any other, whether its sides are so small that only a microscope could show them, or so large that they reach the farthest fixed star. Clearly there can be an infinite number of isosceles triangles just in the length of the equal sides, and each of these can be varied infinitely again by widening or narrowing the angle between those sides. So the number of possible isosceles triangles is infinitely infinite, and yet we are asked to believe of this unimaginable number of objects what we have proved of only one single specimen. This might look like the most extreme Imperfect Induction possible, and yet everyone agrees that it gives us truly certain knowledge. We know, with as much certainty as any knowledge can have, that if we imagine lines drawn from the earth to two stars at equal distances from it, those lines will make equal angles with the line joining the two stars. And we can never have tried the experiment.

The generality of this geometrical reasoning plainly depends on how certainly we know that all isosceles triangles resemble each other exactly. The proposition we proved does not in fact apply to a triangle unless that triangle agrees with our specimen in every quality the proof depends on. The actual length of any side, and the actual size of the angle between any two sides, were not things the proof depended on. They were purely accidental circumstances. So we are perfectly free to apply to every new case of an isosceles triangle what we learn from one case. The whole vast body of certain knowledge in the mathematical sciences rests on similar ground: not only all the truths of geometry, but all the general truths of algebra. We showed in Lesson 7, for instance, that if a and b are two quantities and we multiply their sum by their difference, we get the difference of the squares of a and b. However often we try this, we will find it true. If a = 10 and b = 7, the product of the sum and difference is 17 × 3 = 51; the squares of the quantities are 10 × 10, or 100, and 7 × 7, or 49, and their difference is also 51. But however often we tried the rule, we would not add any certainty to it. When it was proved algebraically, there was no condition limiting the result to particular numbers, so a and b could be any numbers whatever. This generality of algebraic reasoning, which proves a property of infinitely many numbers at once, is one of the chief advantages of algebra over arithmetic. Algebra also has a process called Mathematical Induction, or Demonstrative Induction, which shows off the power of reasoning in a very striking way. Here is a good example. Take the first two odd numbers in a row, 1 and 3, and add them: the sum is 4, exactly two times two. Take three of them, 1 + 3 + 5, and the sum is 9, exactly three times three. Take four, 1 + 3 + 5 + 7, and the sum is 16, exactly four times four. In general, if we take any number of terms of the series 1 + 3 + 5 + 7 + ..., the sum equals the number of terms multiplied by itself. Anyone who knows a very little algebra can prove that this remarkable law is true universally, like this. Let n be the number of terms, and suppose for a moment that the law holds up to n terms, so that—

$$1+3+5+7+\dots +(2n-1)=n^2.$$ Now add $2n+1$, the next odd number in the series, to each side of the equation. It follows that—

$$1+3+5+7+\dots +(2n-1)+(2n+1)=n^2+2n+1.$$ But the last quantity $n^2+2n+1$ is exactly equal to ${(n+1)}^2$. So if the law is true for n terms, it is also true for n + 1 terms. This lets us argue from each single case of the law to the next one. And we have already shown that it is true for the first few cases, so it must be true for all of them. No amount of labor could let a person find out by trial what the first billion odd numbers add up to (a billion here meaning, as the British counted it, a million million). Yet by symbols, that is, by general reasoning, we know for certain that they would add up to a billion billion, not one more or one less. This process of Mathematical Induction is not quite the same as Geometrical Induction, because each case depends on the one before it. But the proof rests on an equally narrow basis of experience, and it produces knowledge that is just as certain and just as general.

Mathematical truths like these depend on observing a few cases, but they get their certainty from our perception that one case is exactly similar to another, so that we believe without any doubt that what is true of one case is true of another. It is very instructive to set beside these cases some others where we have the same kind of ground in observation, but not the same tie of similarity. It was once believed that if you took any whole number, multiplied it by itself, added the number itself, and then added 41, the result would be a prime number, that is, a number that cannot be divided by any whole number except itself and one. In symbols, $$x^2+x+41=\mathrm{\text{prime number.}}$$

This was believed purely on the strength of trial and experience, and it certainly holds for a great many values of x. When x is given in turn each of the numbers in the first line below, the expression $x^2+x+41$ gives the values in the second line, and they are all prime numbers:

But no one could give any reason why it should always be true, and in fact the rule does not always hold. It fails when x = 40. Then we get 40 × 40 + 40 + 41 = 1681, and this is plainly equal to 41 × 40 + 41, or 41 × 41, which is not a prime number.

In the branch of mathematics that deals with the special properties and kinds of numbers, other propositions resting only on observation have been claimed to be always true. Fermat, for example, believed that $2^{2^x}+1$ always gives a prime number, but he could give no reason for the claim. It does in fact hold until the result reaches the large number 4294967297, which was found to be divisible by 641, and that disproved the general statement.

So we find that in some cases a single instance proves a general and certain rule, while in others a very great number of instances is not enough to give any certainty at all. Everything depends on our perception of similarity, or identity, between one case and another. We can see no similarity among all prime numbers that assures us that because one is given by a certain formula, another will be too. But we do find that kind of similarity among sums of odd numbers, and among isosceles triangles.

Exactly the same considerations apply to inductions in physical science. A chemist analyzes a few grains of water and finds that every 9 parts of water contain exactly 8 parts of oxygen and 1 of hydrogen. He feels entitled to assert that the same is true of all pure water, wherever it came from and whatever part of the world it was taken from. But if he analyzes a piece of granite, or a sample of seawater from one part of the world, he has no confidence that it will exactly resemble a piece of granite or a sample of seawater from another part of the earth. So he does not venture to assert of all granite, or all seawater, what he finds true of one sample. Wider experience shows that granite varies a great deal in composition, but that seawater is kept fairly uniform by the constant mixing of currents. Only experience could tell us, in these cases, how far we can safely assert of one sample what we have found out about another. But we have reason to believe that chemical compounds are by nature fixed and unchanging in composition, according to Dalton's laws of combining proportions. No a priori reasoning from the principles of thought could have told us this. We learn it only through wide experiment. But once we have shown it to be true of certain substances, we don't need to repeat the trial with every other substance, because we have every reason to believe that this is a natural law in which all chemical substances resemble each other. A single accurate analysis of a given fixed compound is all we need, then, to tell us the composition of every other portion of the same substance.

We must be careful to notice, however, that all inductions in physical science are only probable, or, if they are certain, the certainty they have is only hypothetical. Can I be absolutely certain that all water contains one part of hydrogen in nine? I am certain only on two conditions:—

1. That this really was the composition of the sample I tested.

2. That any other substance I call water exactly resembles that sample. But even if the first condition is beyond doubt, I cannot be certain of the second. After all, how do I know what water is, except by its being a transparent liquid that freezes into a solid and evaporates into steam, that has a high specific heat, and that has a number of other distinct properties? And can I be absolutely certain that every liquid with all these properties is water? In practice I can be certain, but in theory I cannot. Two substances might have been created so like each other that we have never yet discovered the difference, and then we might constantly be misled by assuming of one what is true only of the other. That this should ever happen with substances that have qualities as distinctive as water's is extremely improbable. But in other cases it is so far from impossible, or even improbable, that it has often happened. Most of the new elements discovered in recent years were, without doubt, mistaken for other elements before. Caesium and Rubidium had long been mistaken for each other, and for Potassium, until Bunsen and Kirchhoff told them apart by means of the spectroscope. Since they are now known to be widely distributed, though in small quantities, it is certain that what was taken for Potassium in many thousands of analyses was partly made up of different substances. Selenium had probably been confused with Sulfur, and there are certain metals that have only recently been told apart: for instance Rhodium, Ruthenium, Iridium, Osmium, and Beryllium; Yttrium, Erbium, Cerium, Lanthanum, and Didymium (which was itself later found to be a mixture of two elements); and Cadmium and Indium. The progress of science will no doubt show that we are mistaken in many of our identifications, and the various difficulties that arise from them will eventually be explained.

Now take a very different case of induction. Are we certain that the sun will rise again tomorrow morning, as it has risen for many thousands of years, and probably for some hundred million years? We are certain only on one condition, or hypothesis: that the planetary system goes on tomorrow as it has gone on for so long. Many causes might exist that could upset all our calculations at any moment. Our sun is believed to be a variable star, and for all we know it might at any moment suddenly explode or flare up, as certain other stars have been seen to do, and then we would all be turned into thin glowing vapor in an instant. It is not at all impossible that a collision once happened in the planetary system, and that the tiny planets, or asteroids, are the result. Even if there is no large meteor, comet or other body able to break up the earth by collision, the sun is probably moving through space at nearly 300 miles a minute, and if some other star should come toward us at a similar speed, the consequences would be terrible beyond imagining. It is very unlikely, however, that such an event would happen even in the course of a million years.

The reader will now see that no mere Imperfect Induction can give certain knowledge. All inference goes forward on the assumption that new instances will exactly resemble old ones in every material circumstance. But with natural phenomena this is purely hypothetical, and we may constantly find ourselves wrong. In Mathematical Induction the certainty came from the cases being hypothetical by their very nature, or being constructed so as to match the conditions exactly. We cannot assert that any triangle existing in nature has two equal sides or two equal angles; in practice it is even impossible for any two lines or angles to be absolutely equal. But it is true all the same that if the sides are equal, the angles are equal. So all certainty of inference is relative and hypothetical. Even in the syllogism, the certainty of the conclusion rests only on the hypothesis that the premises are certain. In fact, it is probable that all reasoning comes down to a single type: what is true of one thing will be true of another thing, on condition that the two exactly resemble each other in every material circumstance.

The reader will now easily understand the nature of reasoning by analogy. Strictly speaking, an analogy is not an identity of one thing with another, but an identity of relations. Take numbers: 7 is not identical with 10, nor 14 with 20, but the ratio of 7 to 10 is identical with the ratio of 14 to 20, so there is an analogy between these numbers. Multiplying two by two is not the same thing as building a square on a line two units long. But there is this analogy between them: the square will contain just as many units of area as there are units in the product of two times two. This analogy is so obvious that we confidently assert that a square mile contains 1760 × 1760 square yards, without testing it at all. In ordinary language, though, analogy has come to mean any resemblance between things that lets us believe of one what we know of the other.

The planet Mars, for example, has an atmosphere with clouds and mists much like our own. It has seas, which can be told from the land by their greenish color, and polar regions covered with snow. (So astronomers of the 1870s read what their telescopes showed them.) The red color of the planet seems to come from its atmosphere, like the red color of our sunrises and sunsets. So much about the surface of Mars is like the surface of the Earth that we readily argue there must be inhabitants there, as there are here. But all we can say for certain is that if the circumstances really are similar, and similar seeds of life have been created there as here, there must be inhabitants. The fact that many circumstances are similar increases the probability. Between the Earth and the Sun, however, the analogy is far fainter. We do speak of the sun's atmosphere as subject to storms and full of clouds, but those clouds are probably heated beyond the temperature of our hottest furnaces. If they produce rain, it must be like a shower of melted iron. And sunspots are disturbances of such tremendous size and nature that the earth, along with half a dozen of the other planets, could easily be swallowed up in one of them. So there is plainly little or no analogy between the Sun and the Earth, and that is why we find it hard to form any idea of what goes on in a sun or star.

Footnote: Lockyer's Elementary Lessons in Astronomy, § 108.

Argument from analogy can be defined as direct inductive inference from one instance to any similar instance. As Mr. J. S. Mill says, it can be reduced to the following formula:—

"Two things resemble each other in one or more respects; a certain proposition is true of the one; therefore it is true of the other." This is no doubt the type of all reasoning, and the certainty of the process depends entirely on how far the cases resemble, or are identical with, each other. In geometry the cases are, by hypothesis, absolutely identical in every material point, and no doubt attaches to the inference. In physical science the identity is a matter of probability, and the conclusion is probable to the same degree. I should add that Mr. Mill holds that Geometrical and Mathematical Induction ought not to be called Induction at all, for reasons whose force entirely escapes me; but the reader will find his views in the 2nd chapter of the 3rd book of his System of Logic.

One form of analogical, or inductive, argument is the constant use of examples and instances. The best way to describe the nature of a class of things is to present one of the things itself, and to point out which of its properties belong to the class, as distinct from those peculiar to that one thing. Throughout these Lessons, as in every book on logic, instances of propositions, of compound or complex sentences, of syllogisms and so on are used again and again, and the reader is asked to apply to all similar cases what he observes in the examples given. It is assumed that the writer chooses examples that truly display the properties in question.

All inductive and analogical inferences rest on the same principles, but their sources of probability differ widely. In analogy we have two cases that resemble each other in a great many properties, and we infer that some further property found in one will probably be found in the other. The very narrow basis of experience is made up for by the high degree of similarity. In the processes more usually treated under the name of Induction, the things generally resemble each other in only two or three properties, and we need more instances to assure us that what is true of these is probably true of all similar instances. In short, the smaller the intension of the resemblance, the greater the extension of our inquiries must be.

In the following Lessons we go on to the ordinary processes of Induction.

Mr. Mill's System of Logic, Book III. Chap. XX. Of Analogy. Mansel's Aldrich, App. Note H. On Example and Analogy.

Questions and Exercises

1. What gives reasoning in geometry its certainty and generality?

2. Find other instances of certain and general reasoning about the properties of numbers.

3. Why are inductive conclusions about prime numbers uncertain and not general?

4. Why is a single instance sometimes enough to justify a universal conclusion, while in other cases the greatest possible number of agreeing instances, without a single exception, is not enough to justify one?

5. What are the strict and the ordinary meanings of the word analogy?

6. Explain the use of examples.

7. Explain exactly the difference between analogical argument and ordinary induction.

Lesson 27: Observation and Experiment

It is safe to say that all knowledge must ultimately rest on experience. "Experience" is simply a general name for all the different feelings and impressions the mind receives at any time in its existence. The mind never creates completely new knowledge independent of experience. All our reasoning powers can do is work out the full meaning of the facts we already have. In earlier centuries, people of the very highest ability believed that the mind, by its own power alone and by thinking hard enough, could discover what the things outside us ought to be, and would turn out to be when examined. They thought we could anticipate Nature, working out from the human mind an idea of what the Creator would have made. The famous philosopher Descartes, for example, held that whatever the mind can clearly conceive may be taken as true. But we can conceive of mountains of gold or oceans of fresh water, and in fact they do not exist. Anything we can clearly conceive must agree with the laws of thought, and so, as far as our intellect is concerned, its existence is not impossible. But the shapes and sizes and ways in which it has pleased the Creator to make things, in this or any other part of the universe, cannot possibly be anticipated by the very limited wisdom of the human mind. They can only be learned by actually examining the things that exist.

In the second half of the 13th century the great Roger Bacon taught clearly, in England, that experience is supremely important as the basis of knowledge. By a curious coincidence, the same doctrine was taken up again in the 17th century by the great Lord Chancellor Francis Bacon, and it has been called the Baconian Philosophy after him. I believe that Roger Bacon was an even greater man than Francis, though Francis is the better known. But the words in which Francis Bacon proclaimed the importance of experience and experiment deserve to be remembered forever. At the beginning of his great work, the Novum Organum, or New Instrument, he points out our proper position as learners in the world of Nature:

"Man, the Servant and Interpreter of Nature, can do and understand as much as he has observed concerning the order of nature in outward things or in the mind; beyond this he can neither know nor do."

That is the first of the aphorisms, or short numbered paragraphs, with which the Novum Organum begins. In the second aphorism he says that the mind on its own can do little and is liable to go wrong. It needs help, in the form of a definite logical method, and supplying that method was the purpose of his New Instrument. The 3rd and 4th aphorisms must be given in full. They are:

"Human science and human power coincide, because not knowing a cause deprives us of the effect. For Nature is not conquered except by obeying her; and what we discover as a cause when we contemplate becomes a rule when we act."

"Man can do nothing for himself except move natural bodies toward or away from each other; Nature working within does the rest."

It would be impossible to express more clearly and completely how we discover science by interpreting the changes we observe in Nature, and then turn that knowledge to useful purposes in the arts and manufacturing. We cannot create or destroy a single particle of matter. It is now known that we cannot even create or destroy force (in today's terms, energy). Nor can we really alter the inner nature of any substance we deal with. All we can do is observe carefully how one substance acts on another by its natural powers, and then, by bringing them together at the right time, accomplish what we want. As Bacon says, "Nature working within does the rest." If it had not been the nature of heat, applied to water, to produce steam with elastic power (the power to push back when compressed), the steam engine obviously could never have been made. The invention of the steam engine came from noticing how useful the force of steam was, and putting it to work accordingly. It is in this sense that the Roman poet Virgil called that person happy who knows the causes of things:

	Felix qui potuit rerum cognoscere causas,

and that Bacon said, "Knowledge is Power." To the extent that we have observed how things happen in Nature, and on what occasions particular effects come about, we can avoid those effects or make use of them as we wish. We do this not by altering the natures of things, but by letting them show their own proper powers at suitable times and in suitable circumstances. In this way, as Tennyson has put it so well, we

	"Rule by obeying Nature's Powers."

Inductive logic deals with the methods of reasoning by which we can successfully interpret Nature and learn the natural laws that different substances obey in different circumstances. In this lesson we consider the first requirement of induction: the experience, or examination of Nature, that supplies us with facts. We get this experience either by observation or by experiment. To observe is simply to notice events and changes produced in the ordinary course of Nature, without being able to control or vary them, or at least without trying to. The early astronomers, for example, observed the motions of the sun, moon and planets among the fixed stars, and gradually detected many of the laws, or regular returns, of those bodies. In the same way the meteorologist observes the ever-changing weather. He notes the height of the barometer, the temperature and humidity of the air, the direction and force of the wind, and the height and character of the clouds, without being able to control any of these things in the least. The geologist, again, is generally a simple observer when he investigates the nature and position of rocks. The zoologist, the botanist and the mineralogist usually rely on mere observation when they examine animals, plants and minerals in the natural condition in which they find them.

In experiment, by contrast, we vary the combinations of things and circumstances as we choose, and then observe the result. This is how the chemist discovers the composition of water: he uses an electric current to separate it into its two constituents, oxygen and hydrogen. The mineralogist may use experiment when he melts two or three substances together to find out how a particular mineral might have been formed. Even the botanist and the zoologist are not limited to passive observation. By moving animals or plants to different climates and different soils, and by what is called domestication, they can test how far the natural forms and species can be altered.

Clearly, where experiment can be used at all, it is the most powerful and direct way of getting facts. We might have to wait years or centuries to come across, by accident, facts that we can easily produce at any moment in a laboratory. Most of the chemical substances now known, and many extremely useful products, would probably never have been discovered at all if we had waited for Nature to present them to us of her own accord. Many forces and changes, too, may be going on in Nature all the time, but so faintly that they escape our senses, and we need some experimental means to detect them. Electricity is no doubt at work in every particle of matter, perhaps at every moment. Even the ancients could not help noticing its action in the lodestone (a natural magnet), in lightning, in the Aurora Borealis (the northern lights), and in a piece of rubbed amber (in Greek, electrum, from which "electricity" takes its name). But in lightning, electricity was too intense and dangerous to study; in the other cases it was too weak to be properly understood. The science of electricity and magnetism could only advance once we had regular supplies of electricity from the ordinary electric machine (a friction generator) or the galvanic battery, and could make powerful electromagnets. Most, if not all, of the effects electricity produces must go on in Nature too, but far too faintly to be observed.

Experiment is also indispensable for another reason. On the surface of the earth we usually meet substances under certain uniform conditions, so observation alone could never teach us what those substances would be like under other conditions. Carbonic acid (carbon dioxide), for example, is met with only as a gas, given off when carbon burns. But when it is exposed to extreme pressure and cold, it condenses into a liquid, and can even be turned into a solid, like snow. Many other gases have been liquefied or solidified in the same way, and there is reason to believe that every substance can take all three forms, solid, liquid and gas, if only the conditions of temperature and pressure can be varied enough. Mere observation of Nature would have led us to the opposite belief: that nearly all substances were fixed in a single state, and could not be turned from solid into liquid, or from liquid into gas.

We must not suppose, however, that we can draw a precise line between observation and experiment and say where one ends and the other begins. The difference is one of degree rather than of kind. All we can say is that the more we vary the conditions artificially, the more we are using experiment. I have said that meteorology is a science of almost pure observation. But if we deliberately climb mountains to observe how the air becomes thinner and colder with height, or go up in balloons for the same purpose, as Gay-Lussac and Glaisher did, we vary our way of observing so much that it becomes almost experimental. Astronomers, again, can almost be said to experiment rather than merely observe when they set up instruments at the same moment as far north and as far south on the earth's surface as possible, to observe how far the position of Venus appears to shift as it crosses the face of the sun in a transit. In this way they can compare the distances of Venus and the sun with the size of the earth.

Sir John Herschel has described this difference excellently in his Discourse on the Study of Natural Philosophy. "Essentially they are much alike, and differ rather in degree than in kind; so that perhaps the terms passive and active observation might better express their distinction; but it is, nevertheless, highly important to mark the different states of mind in inquiries carried on by their respective aids, as well as their different effects in promoting the progress of science. In the former, we sit still and listen to a tale, told us, perhaps obscurely, piecemeal, and at long intervals of time, with our attention more or less awake. It is only by after rumination that we gather its full import; and often, when the opportunity is gone by, we have to regret that our attention was not more particularly directed to some point which, at the time, appeared of little moment, but of which we at length appreciate the importance. In the latter, on the other hand, we cross-examine our witness, and by comparing one part of his evidence with the other, while he is yet before us, and reasoning upon it in his presence, are enabled to put pointed and searching questions, the answer to which may at once enable us to make up our minds. Accordingly it has been found invariably, that in those departments of physics where the phenomena are beyond our control, or into which experimental enquiry, from other causes, has not been carried, the progress of knowledge has been slow, uncertain and irregular; while in such as admit of experiment, and in which mankind have agreed to its adoption, it has been rapid, sure, and steady."

Footnote: Herschel's Discourse, p. 77.

Often, though, Nature has, so to speak, made experiments on a scale and over a length of time that we cannot possibly match. We do not need to test which soil and situation suit a given plant best, for example. We only have to look around and notice the habitat, or place, where it grows naturally in the most flourishing condition, and we can be sure that this shows the result of ages of natural experiment. We would probably never have known the distances of the fixed stars if the earth, by traveling in an orbit 182,000,000 miles across, did not give us a kind of experimental base line for observing them. Because of it we see the stars in very slightly different positions, and so can judge their distances compared with the size of the earth's orbit. Eclipses, transits, occultations (when the moon or a planet passes in front of a star) and remarkable conjunctions of the planets are also kinds of natural experiment. They were often recorded in early times, and so give us data of the greatest value.

Footnote: See Lockyer's Elementary Lessons in Astronomy, Nos. XLVI, XLVII.

Logic can do little or nothing to make someone a sharp or accurate observer. No definite rules can be laid down on the subject. Observing well is an art that can only be gained by practice and training, and one of the greatest advantages of studying the Natural Sciences is that it develops the ability to observe clearly and steadily. Logic can, however, give us one caution, which Mr. J. S. Mill has pointed out well: we must distinguish accurately between what we really observe and what we only infer from the facts we observe. As long as we only record and describe what our senses have actually witnessed, we cannot make a mistake. The moment we assume or infer anything, we can go wrong. For example, suppose we examine the sun's surface through a telescope. We observe that it is intensely bright except where there are small breaks, or round openings, in the surface with a dark interior. We are irresistibly led to conclude that the inside of the sun is colder and darker than the outside, and we record as a fact that we saw the dark interior of the sun through certain openings in its luminous atmosphere. But that record would contain a mistaken inference. We saw nothing but dark spots, and as observers we should have done no more than record the shape, size, appearance and changes of those spots. Whether they are dark clouds above the luminous surface, or glimpses of the dark interior, or, as is now (in the 1870s) almost certainly inferred, something entirely different from either, can only be proved by comparing many unprejudiced observations.

The reader cannot keep in mind too often this caution against confusing the facts observed with the inferences drawn from them. It is not too much to say that nine-tenths of what we seem to see and hear is inferred, not actually felt. Every sense has what are called acquired perceptions: the power, built up by long experience, of judging unconsciously many things that cannot be perceived directly. The eye cannot see distance. Yet we constantly imagine, and say, that we see things at such and such a distance, without realizing that this is the result of judgment. As Mill remarks, it is saying too much to say "I saw my brother." All I positively know is that I saw someone who closely resembled my brother, as far as I could observe. It is only by judgment that I can say he was my brother, and that judgment might be wrong.

Nothing is more important in observation and experiment than not letting any prejudice or theory influence us, so that we record the facts observed correctly and give them their proper weight. Someone who fails to do this will almost always be able to find facts that support an opinion, however mistaken. For example, most uneducated people still believe firmly that the moon has a great influence on the weather. The changes of the moon, full, new and half moon, happen four times every month, and it is supposed that any change can influence the weather at least on the day before or the day after it. So there will be twelve days out of every 28 on which any change in the weather would be put down to the moon, and over a year many changes will probably be recorded in this way as supporting the opinion. The uneducated observer is struck by these cases and remembers them carefully. But he fails to notice, or at least to remember, that changes of weather also often happen when there is no change of the moon at all. The question could only be settled by a long course of careful and unbiased observation, in which all the facts, favorable and unfavorable, were recorded equally. All the observations that have been published go against the idea that the moon has any such influence as the popular mind gives it.

At the same time, it would be a mistake to think that the best observer or experimenter is the one who holds no previous opinions or theories about the subject he is investigating. On the contrary, the great experimenter is the one who always has a theory, or even a whole crowd of theories or ideas, in his mind, but is always putting them to the test of experience and throwing out the ones that are false. The number of things that could be observed and experimented on is infinite. If we simply set to work recording facts without any definite purpose, our records will be worthless. We need some opinion or theory to guide our choice of experiments, and we are more likely to hit on the truth that way than by working at random. But the great requirement of the true scientist is that he be completely unbiased, and give up any opinion as soon as facts are observed that are inconsistent with it.

The famous Turgot (the French economist and statesman) put it well: "the first thing is to invent a system; the second thing is to be disgusted with it." That is, we ought to have some idea of the truth we are looking for, but we should immediately put it to a severe trial, as if we were inclined to distrust and dislike it, rather than be biased in its favor. Probably few people have entertained more false theories than Kepler and Faraday; few have discovered or established truths of greater certainty and importance. Faraday himself has said:

"The world little knows how many of the thoughts and theories which have passed through the mind of a scientific investigator, have been crushed in silence and secrecy by his own severe criticism and adverse examination; that in the most successful instances not a tenth of the suggestions, the hopes, the wishes, the preliminary conclusions have been realized."

Footnote: Modern Culture edited by Youmans, p. 222. [Macmillan and Co.]

The student is strongly advised to read Sir John Herschel's Preliminary Discourse on the Study of Natural Philosophy (Lardner's Cabinet Cyclopædia), especially Part II, Chapters 4 to 7, on Observation, Experiment, and the Inductive Processes generally.

Questions and Exercises

1. What is the false method of science that Bacon protested against?

2. Explain exactly what Bacon meant when he said that man is the Servant and Interpreter of Nature, and that Knowledge is Power.

3. How does experiment differ from observation?

4. Classify the sciences according to whether they use passive observation, experiment, or both.

5. Name the chief ways in which experiment is better than mere observation.

6. What is the chief precaution needed in observation?

7. Explain how it is possible to anticipate Nature and yet base all our conclusions on the results of experience.

Lesson 28: Methods of Induction

We now have to look at the methods that can be laid down to guide us when we search for general truths, or laws of nature, among the facts we have gathered by observation and experiment. Induction means inferring from particulars to generals: spotting a general truth among the particular cases in which it shows up. In physical science, though, the truths to be discovered are usually about the connection of cause and effect, and we generally call them laws of causation or natural laws. By the Cause of an event we mean the circumstances that must have come before it in order for the event to happen. And it is usually impossible to say that an event has one single cause and no more. Producing an effect normally takes many different things, conditions or circumstances, and all of them must be counted as causes, or as necessary parts of the cause. Take the loud bang of a gun going off. Its cause is not simply the pulling of the trigger, which is only the last visible cause, or the occasion, of the explosion. The qualities of the powder, the right shape of the barrel, a charge in the barrel to resist the blast, the proper arrangement of the percussion cap and the powder, and the air surrounding the gun are all among the circumstances needed for that loud report. If any one of them had been missing, it would not have happened.

Again, the cause of water boiling is not just heating it up to a certain temperature. The vapor must also be able to escape once it reaches a certain pressure. In the same way, the freezing of water does not depend merely on drawing off heat until the temperature falls below 0° Centigrade (Celsius). So the work of Induction is to detect the circumstances that will always produce a given effect. As soon as we know them, we have a law, or uniformity of nature, of greater or less generality.

In this lesson and the ones that follow I will often need, besides cause and effect, the words antecedent and consequent, and the reader should notice what they mean. An antecedent is any thing, condition or circumstance that exists before an event or phenomenon, or possibly at the same time as it. A consequent is any thing, circumstance, event or phenomenon that is different from all the antecedents and follows after they have come together. An antecedent is not necessarily a cause, because the effect might have happened without it. The sunlight, for example, may be an antecedent of a house burning down, but it is not the cause, because the house would burn just as well at night. A necessary, or indispensable, antecedent, however, is the same thing as a cause: it is that without which the effect would not happen.

I will also often use the word phenomenon. It simply means anything that appears, and so is observed by the senses. The word comes from the Greek φαινόμενον (phainomenon), "that which appears," and this origin matches its use in logic exactly.

The first method of Induction is the one Mr. J. S. Mill has aptly called the Method of Agreement. It rests on this rule: "If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon." I think the meaning of this First Canon of inductive inquiry could be put more briefly: the sole invariable antecedent of a phenomenon is probably its cause.

To apply this method, we collect as many instances of the phenomenon as we can and compare their antecedents. The causes will be somewhere among them. But if we notice that certain antecedents are sometimes present and sometimes absent without seeming to affect the result, we conclude that they cannot be necessary antecedents. So it is the one antecedent, or group of antecedents, that is always present when the effect follows that we take to be the cause. Here is an example. Bright rainbow colors appear on bubbles, on films of tar floating on water, on thin sheets of mica, on cracks in glass, and between two pieces of glass pressed together. When we examine all these cases, they seem to agree in nothing except the presence of a very thin layer or plate, and it seems to make no noticeable difference what kind of matter the plate is made of, whether solid, liquid or gas. So we conclude that these colors are caused simply by the thinness of the plates, and the theory of the interference of light proves this conclusion true. Sir David Brewster gave a beautiful proof, along the same lines, that the colors seen on mother-of-pearl are caused not by the nature of the substance but by the shape of its surface. He took impressions of mother-of-pearl in wax and found that, although the substance was now entirely different, the colors were exactly the same. It was later found that a metal plate whose surface was marked with very fine, close-set grooves would show iridescent colors like those of mother-of-pearl. So it is clear that the shape of the surface, the only invariable antecedent or condition needed to produce the colors, must be their cause.

The Method of Agreement runs into a serious difficulty, which Mill calls the Plurality of Causes: the same effect may be due to different causes in different instances. If we look carefully into the cause of heat, for example, we find it is produced by friction, by burning or combustion, by electricity, by pressure and so on. So even if one and the same thing happened to be present in every case we examined, it would not follow that this thing was the cause. The second method of induction, which we will look at now, is free from this difficulty. It is known as the Method of Difference, and Mill's Second Canon states it like this:

"If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former; the circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon."

In other words: the antecedent that is always present when the phenomenon follows, and always absent when the phenomenon is absent, with the other circumstances staying the same, is the cause of the phenomenon in those circumstances.

This lets us prove clearly that friction is one cause of heat. When two sticks are rubbed together they get hot; when they are not rubbed, they do not. Sir Humphry Davy showed that even two pieces of ice rubbed together in a vacuum produce heat, as their melting shows, and so he completely demonstrated that the friction is the source and cause of the heat. We prove that air is what carries sound to our ears by striking a bell inside the receiver (the glass jar) of an air pump, as Hawksbee first did in 1705. When the receiver is full of air we hear the bell; when it holds little or no air we do not. We learn that sodium, or any of its compounds, produces a spectrum with a bright yellow double line by noticing two things: there is no such line in the spectrum of a light when no sodium is present, but if the smallest quantity of sodium is thrown into the flame or other source of light, the bright yellow line appears instantly. Oxygen is the cause of breathing and life, because if an animal is put into a jar of ordinary air from which the oxygen has been removed, it soon suffocates.

This is essentially the great method of experiment. How useful it is depends mainly on one precaution: varying only one circumstance at a time and keeping all the others exactly as they were. Thomson and Tait put this into one of the rules for conducting experiments in their great treatise on Natural Philosophy, Vol. I. p. 307:

"In all cases when a particular agent or cause is to be studied, experiments should be arranged in such a way as to lead if possible to results depending on it alone; or, if this cannot be done, they should be arranged so as to increase the effects due to the cause to be studied till these so far exceed the unavoidable concomitants, that the latter may be considered as only disturbing, not essentially modifying the effects of the principal agent."

Here is what an imperfect and unsatisfactory experiment would look like. Suppose we took air whose oxygen had been turned into carbonic acid (carbon dioxide) by burning carbon in it, and argued that, because an animal dies in that air, oxygen is the cause of breathing. Instead of simply removing the oxygen, we have brought in a new substance, carbonic acid, which is perfectly capable of killing the animal by its own poisonous properties. In fact the animal would suffocate even if a good deal of oxygen were left. So the presence of the carbonic acid is a disturbing circumstance that confuses the experiment and spoils it.

We can prove that the force of gravity exists, and even measure how strong it is, by delicately hanging up a small ball about the size of a marble and then suddenly bringing a very heavy lead ball, weighing a ton or more, close to it. The small ball will be attracted and set moving. But the experiment would be worthless unless it were done with the greatest care. Obviously, moving the big lead ball suddenly would disturb the air, shake the room, set up air currents by being colder or warmer than the air around it, and even cause electrical attractions or repulsions. All of these would probably disturb the small ball far more than the force of gravitation does.

As Sir John Herschel has pointed out, beautiful examples of experiment by this method can be found in the research by which Dr. Wells discovered the cause of dew. On a clear, calm night, stretch a sheet or some other covering a foot or two above the ground, so that it screens the ground beneath it from the open sky. Dew will form on the grass around the screen but not underneath it. Since the temperature, the moisture of the air and the other circumstances are exactly the same, the open sky must be an indispensable antecedent of dew. Nature in fact tries the same experiment for us. If we observe the dew on two nights that differ in nothing except that one is cloudless and the other cloudy, we will find that a clear open sky is needed for dew to form.

Often we cannot apply the Method of Difference perfectly, by varying only one circumstance at a time. For example, we generally cannot test the qualities of the same substance as a solid and as a liquid without any other change in the circumstances, because to melt it or freeze it we have to change its temperature. The temperature might then be the cause of what we put down to the substance being liquid or solid. In cases like this we have to fall back on what Mill calls the Joint Method of Agreement and Difference. It consists of applying the Method of Agreement twice: first to a number of instances where an effect is produced, and second to a number of quite different instances where the effect is not produced. It must be clearly understood, though, that the negative instances differ from the positive ones in several circumstances. If they differed in only one, we could use the simple Method of Difference. Iceland spar, for instance, has the curious power of making things seen through it look double. This phenomenon, called double refraction, belongs to many other crystals as well. We could prove at once that it is due to crystalline structure if we could get some transparent substance in both a crystallized and an uncrystallized state, with no other change at all. We can't, but we do have a fairly satisfactory proof. We observe that uniform, transparent, uncrystallized substances agree in not having double refraction, and that crystalline substances, on the other hand, agree in having that power, apart from certain exceptions that are easy to explain. The principle of the Joint Method can be stated in the following rule, which is Mill's Third Canon:

"If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance; the circumstance in which alone the two sets of instances (always or invariably) differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon."

I have put in the words in parentheses myself, because without them the canon seems to me to say exactly the opposite of what Mill means.

It may help in understanding these inductive methods exactly if I give a symbolic picture of them, in the way Mill does. Let A, B, C, D, E and so on be antecedents that can be combined in various ways, and let a, b, c, d, e and so on be the effects that follow from them. Suppose we can collect the following sets of antecedents and effects:

Then we can apply the Method of Agreement, and little doubt will remain that A, the sole invariable antecedent, is the cause of a.

The Method of Difference is represented well enough by this:

Here B and C stay completely unchanged, and we find that the presence or absence of A brings about the presence or absence of a. So A is the cause of a when B and C are present. But the reader should be warned against thinking that this proves A is the cause of a under every circumstance whatever.

The Joint Method of Agreement and Difference is represented in the same way:

Here the presence of A is followed by a, as in the simple Method of Agreement, and the absence of A, in circumstances different from the earlier ones, is followed by the absence of a. So it is very highly probable that A is the cause of a. But it is easy to see that A is not the only circumstance in which the two sets of instances differ; if it were, we could apply the simple Method of Difference to any pair of them. What we can say is that the presence of A is a circumstance in which one set invariably, or uniformly, or always, differs from the other set. So the Joint Method stands in for the simpler Method of Difference in cases where that method cannot properly be put to work.

Herschel's Discourse, part II. chap. 6, p. 144.

Mill's System of Logic, book III. chaps. 8 and 9.

Lesson 29: Methods of Quantitative Induction

The methods of Induction described in the last Lesson dealt only with whether the event whose cause we were looking for happened or did not happen. That is how we learned that friction is one cause of heat. We saw that two solid bodies rubbed together, even two pieces of ice, produced heat, and that when they were not rubbed, no heat was produced. But this is a very elementary sort of experiment. As an investigation moves forward, we always need to measure the exact quantity of an effect, if it is the kind of thing that can be more or less, and to connect it with the quantity of the cause. In fact there is a natural course of progress that we follow in every inquiry of this kind, and it can be laid out as a series of questions:

1. Does the antecedent invariably produce an effect?

2. In what direction is that effect?

3. How much is that effect in proportion to the cause?

4. Is it uniformly in that proportion?

5. If not, according to what law does it vary?

Take, for instance, the way heat changes the size of bodies. The first question is whether heating a solid body, say a bar of iron, changes its length. The simple method of difference lets us answer that it does. The next inquiry shows that almost all substances get longer, or larger in size, when heated, but that a very few, such as India rubber, and water below 4.08° Centigrade (Celsius), get smaller. Next we work out the proportion of the change to each degree of temperature, which is called the coefficient of expansion. Iron, for example, expands by 0.0000122 of its own length for every 1° Centigrade between 0° and 100°.

Still more careful inquiry shows, however, that the expansion is not uniformly proportional to the temperature. Most metals expand faster and faster the hotter they get. But we don't need to go into the details of the subject here.

Or take the fixed stars. They have come up often in these Lessons, but the reader probably knows that they are not really fixed. For any particular star, the astronomer actually has to answer the five questions below.

Firstly. Does the star move?

2ndly. In what direction does it move?

3rdly. How much does it move in a year or a century?

4thly. Does it move uniformly?

5thly. If not, according to what law does the motion vary in direction and speed?

Every science, and every question in a science, starts out as a matter of fact only. Then it becomes a matter of quantity, and little by little it becomes more and more precisely quantitative. Thirty years ago most of the phenomena of electricity and electromagnetism were known only as facts. Now most of them can be exactly measured and calculated.

As soon as phenomena can be measured like this, we can use a further Method of Induction, and a very important one. It is really the Method of Difference again, but used under far more favorable conditions, where every degree and quantity of a phenomenon gives us a new experiment and a new proof of the connection between cause and effect. It may be called the Method of Concomitant Variations, and Mr. J. S. Mill states it like this, in what he calls the Fifth Canon of Induction:

"Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation."

Sir John Herschel states the same method as follows: "Increase or diminution of the effect, with the increased or diminished intensity of the cause, in cases which admit of increase and diminution," and he adds, "Reversal of the effect with that of the cause."

The examples of this method are countless. Mr. Joule, of Manchester, proved beyond doubt that friction is a cause of heat. He spent exact quantities of force (in today's terms, energy) rubbing one substance against another, and showed that the heat produced was greater or less in exact proportion as the force was greater or less. We can also apply the method to many cases that had earlier been handled by the simple method of difference. Instead of striking a bell in a complete vacuum, for instance, we can strike it with a very little air left in the receiver (the glass jar) of the air pump. We then hear a very faint sound, which grows louder or softer every time we increase or decrease the density of the air. This experiment fully satisfies anyone that air is the cause of the transmission of sound.

This method often lets us detect the material connection that exists between two bodies. For a long time it was doubtful whether the red flames seen during total eclipses of the sun belonged to the sun or to the moon. But during the last eclipse of the sun, it was noticed that the flames moved with the sun, and that the moon gradually covered and uncovered them at successive moments of the eclipse. From then on, no one could doubt that they belonged to the sun.

Again, whenever phenomena go through Periodic Changes, increasing and decreasing by turns, we should look for other phenomena that go through changes in exactly the same periods, and there will probably be a connection of cause and effect between them. This is how the tides are proved to be caused by the attraction of the moon and the sun. The periods of high and low tides, and of spring and neap tides, follow one another at intervals that match the apparent revolutions of those bodies around the earth. The moon turns on its own axis in exactly the same period in which it goes around the earth, so that for unknown ages the same side of the moon has always faced the earth. This is a perfect case of concomitant variations, and it proves beyond doubt that the earth's attraction governs the moon's turning on its own axis.

The most extraordinary case of variations, however, is the connection that has been shown in recent years between the Aurora Borealis, magnetic storms, and the spots on the sun. Only in the last 30 or 40 years has it become known that the magnetic compass needle is subject from time to time to very slight but curious movements. At the same moments there are usually natural currents of electricity running in telegraph wires, strong enough to interfere with sending messages. These disturbances are known as magnetic storms, and they are often observed when a fine display of the Northern or Southern Lights is taking place somewhere on the earth. Observations over many years have shown that these storms reach their worst at the end of every eleven years (the maximum falling around the present year, 1870), and then weaken until the next period of eleven years has passed. Close observation of the sun over 30 or 40 years has shown that the size and number of the dark spots, which are gigantic storms going on at the sun's surface, increase and decrease at exactly the same periods as the magnetic storms at the earth's surface. No one can doubt, then, that these strange phenomena are connected, although how they are connected is quite unknown. It is now believed that the planets Jupiter, Saturn, Venus and Mars are the real causes of the disturbances, because Balfour Stewart and Warren de la Rue have shown that the motions of these planets correspond exactly to the periods of the sunspots. This is a most remarkable and far-reaching case of concomitant variations.

Next we have to consider a method of Induction that must be used when several causes act at once and their effects all blend together, producing a joint effect of the same kind as the separate effects. Suppose that in one experiment friction, combustion, compression and electric action are all going on at the same time. Each of these causes will produce a quantity of heat, the quantities will add together, and it will be hard or impossible to say how much is due to each cause separately. We may call this a case of the homogeneous intermixture of effects. The name means that the joint effect is of the same kind as the separate effects. Mr. Mill distinguishes it from cases of the heterogeneous, or, as he calls it, the heteropathic intermixture of effects, where the joint effect is totally different in kind from the separate effects. If we bend a bow too far, for example, it breaks instead of bending further. If we warm ice, it soon stops rising in temperature and melts. If we warm water, it rises in temperature homogeneously for a while, but then it suddenly stops, and an effect of a totally different kind follows: the production of vapor, or possibly an explosion.

Now, when the joint effect is of a heterogeneous kind, the method of difference is enough to find the cause of its occurrence. Whether a bow or a spring will break under a given weight is easy to test, and whether water will boil at a given temperature, at any given reading of the barometer, is just as easy to find out. But in the homogeneous intermixture of effects our task is more complicated. There are several causes, each producing part of the effect, and we want to know how much is due to each one. In this case we must use a further Inductive Method, which Mr. Mill calls the Method of Residues and states in his Fourth Canon:

"Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents."

If we know that the joint effect a, b, c is due to the causes A, B, and C, and we can prove that a is due to A and b to B, it follows that c must be due to C. There can be no simpler case of this than finding the exact weight of a load of goods in a cart. We weigh the cart and load together, and then subtract the tare, the weight of the cart alone, which we found earlier. In the same way we can find how much of the spring tides is due to the attraction of the sun, provided we have already worked out the height of the tide due to the moon, which will be about the average height of the tides over the whole lunar month. Subtract the moon's tide, and what remains is the sun's tide.

Newton used this method in a beautiful experiment to measure the elasticity of different substances. He made balls of the substances, let them swing against each other, and then observed how far they bounced back compared with how far they had originally fallen. But the loss of motion is due partly to imperfect elasticity and partly to the resistance of the air. He measured the second effect in the simplest possible way: he let the balls swing without striking each other, and observed how much smaller each swing was than the one before. This made it easy for him to calculate the amount that had to be subtracted for the resistance of the air.

This is the method we use when we make allowance for errors, or apply the necessary corrections, in observations. Few thermometers are quite accurate. But if we put a thermometer into melting snow, which has a temperature of exactly 0° Centigrade, or 32° Fahrenheit, we can see exactly how far below or above the true point the mercury stands. That tells us how much we should add to or subtract from the thermometer's readings to make them correct. The height of the barometer is affected by several causes besides changes in the pressure of the air. It is lowered by the capillary repulsion between the glass tube and the mercury. It is raised by the expansion of the mercury with heat, if the temperature is above 32° Fahrenheit. And it may be raised or lowered by any error in the length of the scale used to measure the height. In an accurate observation, all these effects are calculated and allowed for in the final result.

In chemical analysis this method is used constantly to find the proportional weights of substances that combine with each other. The composition of water, for example, is found like this. We take a known weight of copper oxide, pass hydrogen over it in a heated tube, and condense the water produced in a tube containing sulfuric acid. Subtracting the original weight of the condensing tube from its final weight tells us how much water was produced. The quantity of oxygen in that water is found by subtracting the final weight of the copper oxide from its original weight. If we then subtract the weight of the oxygen from the weight of the water, we learn the weight of the hydrogen that we have combined with the oxygen. When the experiment is done very carefully, as described in Dr. Roscoe's Lessons in Elementary Chemistry (p. 38), we find that 88.89 parts by weight of oxygen unite with 11.11 parts of hydrogen to form 100 parts of water.

This method is used in every science that allows quantities to be measured, but above all in astronomy, the most exact of all the sciences. Almost all the causes and effects in astronomy have been discovered as residual phenomena. That is, astronomers calculate the effects of all the known attractions on a planet or a satellite, and then observe how far it is from the place those calculations predicted. When this was done very carefully for Uranus, the planet was still found to be sometimes ahead of and sometimes behind its true place. This residual effect pointed to some cause of attraction that was not yet known, and as a result that cause was soon discovered in the shape of the planet Neptune. The motions of several comets have been calculated in the same way, but it is observed that they fail to return at the expected time. There is a discrepancy that points to some obstructing power in the space they pass through, whose nature is not yet understood.

Mill's System of Logic, Book III. Chap. 10, Of the Plurality of Causes; and of the Intermixture of Effects.

Questions on the Methods of Induction

1. Define exactly what is meant by a cause of an event, and distinguish between cause, occasion, and antecedent.

2. Point out all the causes involved in the following phenomena:

(1) The burning of a fire.

(2) The ordinary growth of plants.

(3) The cracking of a glass by hot water.

3. State and explain in your own words Mr. Mill's first three Canons of Inductive Method.

4. Point out exactly how the Joint Method differs from the simple Method of Difference.

5. Give some examples of simple experiments that completely fulfill the conditions of the Method of Difference.

6. What can you infer from the following instances?

7. (1) Friction changes the temperature of the bodies rubbed together.

(2) The sun is thought to move through space.

(3) A ray of light passing into or out of a denser medium is bent aside. Point out the successive questions that would have to be settled in investigating each of the phenomena above.

8. Find some simple examples of the homogeneous and the heterogeneous intermixture of effects, and of the methods of concomitant variations and of residues.

9. Since 1842 there has been a great reform of the British tariff, and a great increase in British trade. Does this coincidence prove that the first circumstance is the cause of the second?

10. Suppose we did not know the causes of the following phenomena. By what methods should we investigate each one?

(1) The connection between the barometer and the weather.

(2) A person poisoned at a meal.

(3) The connection between the hands of a clock.

(4) The effect of the Gulf Stream on the climate of Great Britain.

Lesson 30: Empirical and Deductive Methods

So far we have treated deduction and induction as if they were two entirely separate and independent methods. In reality they are often blended together, or used by turns, in the search for truth. In fact, all the more important and far-reaching investigations in science can be said to rely on one just as much as on the other. Probably the greatest merit of Mr. J. S. Mill's writings on logic is that he shows how completely inadequate the so-called Baconian Method is for detecting the more obscure and difficult laws of nature. Bacon advised that we should always begin by collecting facts, sort them according to how they agree and differ, and gradually gather from them laws of greater and greater generality. He protested against "anticipating nature" altogether, that is, against forming our own hypotheses and theories about what the laws of nature probably are. He seems to have thought that arranging facts systematically would take the place of every other method. The reader will soon see that the progress of science has not borne out his opinions.

When a law of nature has been found purely by induction from certain observations or experiments, and has nothing else to guarantee its truth, it is called an empirical law. As Mill puts it, "Scientific inquirers give the name of Empirical Laws to uniformities which observation or experiment has shown to exist, but on which they hesitate to rely in cases varying much from those which have been actually observed, for want of seeing any reason why such a law should exist." The name comes from the Greek word ἐμπειρία (empeiria), meaning experience or trial. Examples of such laws are plentiful. We learn empirically that a certain strong yellow color at sunset, or an unusual clearness in the air, warns of rain; that a quick pulse is a sign of fever; that horned animals are always ruminants (animals that chew the cud); that quinine does good to the nervous system and to the health of the body in general; and that strychnine has a terrible effect of the opposite kind. We know all of these are true from repeated observation, but we can give no other reason for their truth. That is, we cannot bring them into harmony with any other scientific facts, and we could never have deduced them or predicted them from what we knew before. The connection between sunspots, magnetic storms, auroras and the motions of the planets, mentioned in the last lesson, is perhaps the most remarkable empirical induction known in the 1870s, for no one has yet given any hint of how these magnetic influences could act across the vast distances of the planetary system. The properties of the various alloys of metals are also good examples of empirical knowledge. When two or three metals are mixed in a given proportion for the first time, no one can tell in advance what the mixture will be like. No one could have predicted that brass would be both harder and more ductile than either of the metals it is made from, copper and zinc; that copper, mixed with the very soft metal tin, would make hard, ringing bell-metal; or that a certain mixture of lead, bismuth, tin and cadmium would melt at a temperature (65° centigrade, or 149° Fahrenheit) far below that of boiling water.

Footnote: Roscoe's Lessons in Elementary Chemistry.

However useful empirical knowledge may be, it matters very little compared with the well-connected and fully explained body of knowledge that makes up an advanced, deductive science. In fact, a science becomes perfect exactly in proportion as it becomes deductive and lets us bring more and more facts that seem unconnected under the same law. Whoever knows exactly why a thing happens will also know exactly in which cases it will happen, and what difference in the circumstances will stop it from happening. Take the simple fact that hot water cracks glass. Most people learn this empirically. They have a vague idea that hot water has a natural and unavoidable tendency to break glass, and that thin glass, being more fragile than other glass, will be broken more easily. Physical science, however, gives a very clear reason for the effect. It shows that this is just one case of the general tendency of heat to make substances expand. The crack happens when the heated glass manages to expand in spite of the colder glass it is joined to. But then we see at once that the same will not be true of thin glass vessels. The heat passes through them so quickly that the whole of the glass is heated almost equally. And so chemists regularly use thin, even glass vessels to hold or boil hot liquids, without any fear of the cracks that would be sure to happen in thick glass vessels or bottles.

The history of science would show beyond question that deduction was the key to all the greatest discoveries. Newton, who after Galileo was the chief founder of experimental science, had without doubt the greatest power of deductive thought any human being has ever had. It is striking to compare his results in optics with his results in chemistry, or rather alchemy. It is not widely known that Newton really was an alchemist. He spent days and nights experimenting in his laboratory, trying to discover the secret of turning metals into gold. But in this research everything was purely empirical, and he had no clue to guide him toward successful experiments. A few lucky guesses in his famous Queries (the questions he added at the end of his Opticks) are all that came of this labor. In optics it was quite different. There he grasped general laws, and every experiment led him to design several more and to predict their results, each more beautiful than the last. In this way he was able to lay, beyond all doubt, the foundations of the science of the spectrum, which is now producing such wonderful results. Some people may suppose that Newton, living not long after Bacon, adopted the Baconian method. But I believe there is no mention of Bacon anywhere in Newton's works, and it is certain that he did not use Bacon's method. The Principia, though it appeals to experiment and observation all the way through, is nonetheless the product of a constant, sustained effort of deductive mathematical reasoning.

What Mill has called the Deductive Method, but which I think might more fittingly be called the Combined or Complete Method, consists of using induction and deduction by turns. It can be said to have three steps:

1. Direct Induction.

2. Deduction, or, as Mill calls it, Ratiocination.

3. Verification.

The first step is a rough and simple appeal to experience, enough to give us a glimpse of the laws at work but not enough to prove them true. Taking these laws as true for the time being, we then reason out what their effects would be in other cases. A further appeal to experience then either verifies the laws we assumed or shows them to be false. In short, there are two appeals to experience, joined by reasoning in between. Newton, for example, passed a ray of sunlight through a glass prism and found that it spread out into a band of colors like those of the rainbow. He adopted the theory that white light is really a mixture of lights of different colors, which get separated as they pass through the prism. He saw that if this were true, then if he passed a single ray from the spectrum, say the yellow ray, through a second prism, it ought not to break up into different colors again. It should stay yellow, whatever was done with it afterward. When he tried it, he found that this was so. He then devised a whole series of similar experiments confirming it, which verified his theory beyond all possible doubt.

It was no mere accident that led Pascal to have a barometer carried to the top of the mountain Puy de Dôme in France. Galileo, it is true, learned by accident that water will not rise more than 33 feet in an ordinary pump, and this led him to claim that it was the limited weight of the atmosphere that made the water rise. Torricelli reasoned from this theory. Mercury is fourteen times as heavy as water, so, he saw, it ought not to rise more than one-fourteenth of that height, or about 29 or 30 inches. The experiment was tried, and it verified the theory. But it took the genius of Pascal to see that the experiment needed to be varied in another way, by carrying the mercury barometer to the top of a mountain. If the weight of the atmosphere really was what held the mercury up, the mercury ought to stand lower on the mountain than at the bottom, because only the upper part of the atmosphere would be pressing down on the mountaintop. The experiment succeeded, and completely verified the original hypothesis. The progress of the experimental sciences depends mainly on this way in which one experiment leads to others and brings to light new facts. In all probability we would never have noticed those facts if we had kept to the purely Baconian method of collecting the facts first and doing the induction afterward.

The greatest achievement of the deductive method is nothing less than the theory of gravitation, which is a perfect example of how the method works. Here the first induction was, we may suppose, the famous fall of the apple, which happened while Newton was sitting in an orchard, during the time he had left London because of the Great Plague. The fall of the apple, so we are told, led Newton to reflect that there must be some power tending to pull bodies toward the earth, and he asked himself why, in that case, the moon did not fall onto the earth. The Lancashire astronomer Horrocks had brought another fact to his mind: when a stone tied to a string is whirled around, it pulls on the string with a force often called centrifugal force. Horrocks had remarked that the planets, as they revolve around the sun, must in the same way tend to fly off from the center. Newton knew Horrocks's views, and so he may have been led to suppose that the earth's attractive force might exactly balance the moon's centrifugal tendency, and so keep the moon going around and around.

But it so happened that the world already had certain empirical laws about the motions of the planets, and without them Newton could hardly have gone any further. Kepler had spent a lifetime observing the heavenly bodies and making up hypotheses to explain their motions. Most of his ideas were wild and baseless, but a lifetime of work was rewarded when he established the three laws that bear his name. They describe the shape of the orbits the planets travel in, and the relation between the size of a planet's orbit and the time the planet takes to go around it. Newton was able to show by geometrical reasoning that if one body revolved around another, attracted toward it by a force that decreases as the square of the distance increases, it would necessarily travel in an orbit for which Kepler's laws hold true, and which would therefore be exactly like the orbits of the planets. This was a partial verification of his theory by an appeal to the results of experience. But several other thinkers had already got that far in the subject. Newton's chief claim to honor is that he kept on with his deductions and verifications until he reached complete proof. To do this, he first had to show that the moon really does fall toward the earth just as fast as a stone would if it were in the same position. Using the best figures then available for the distance of the moon, Newton calculated that the moon falls through 13 feet in one minute, but that a stone raised to the same height would fall through 15 feet. Most people would have taken this near agreement as proof of the theory. But Newton's love of certain truth set him apart even from most scientists, and the difference made him lay "aside at that time any further thoughts of this matter."

It was not until many years later (probably fifteen or sixteen) that Newton heard of more exact data for calculating the distance of the moon, and was able to explain the difference. His theory of gravitation was then verified as far as the moon was concerned. But for him this was only the start of a long series of deductive calculations, each ending in a verification. If the earth and the moon attract each other, and so do the sun and the earth, there is no reason why the sun and the moon should not attract each other too. Newton worked out the consequences of this inference, and showed that the moon would not move as if it were attracted by the earth alone, but would go sometimes faster and sometimes slower. Comparison with Flamsteed's observations of the moon showed that this was the case. Newton argued further that since the waters of the ocean are not rigidly fixed to the earth, they might attract the moon, and be attracted by it in return, independently of the rest of the earth. That would cause certain daily motions of the water exactly like the tides, and there were the tides to verify it. What makes Newton stand above every other scientist is the almost superhuman power with which he traced out the consequences of his theory geometrically and tested them again and again against experience.

What he began has been going on ever since. The positions of the moon and planets are calculated for every day on the assumption that Newton's law of gravitation is absolutely true. Every night their positions are observed, as far as possible, at Greenwich or some other observatory. When the observed position is compared with the predicted one, there is always some degree of error, and if the two ever coincided exactly it would only be by accident. The theory is never proved completely true, and never can be. But the more accurately its results are calculated, and the more perfect the astronomer's instruments become, the closer the agreement gets. So the rough observations of Kepler, and the few small facts that worked on Newton's mind, became the foundation of a theory that gave us endless ways of predicting new facts. By constant verification, as far as human accuracy can reach, it has been placed beyond all reasonable doubt.

If there were room, it could be shown that all the other great theories have followed much the same course. The wave theory (undulatory theory) of sound was in fact very nearly verified by Newton himself, although when he calculated the speed of sound from it there was again a difference, which only later research could explain. This theory no doubt suggested the matching theory of light. When Young, Fresnel and others worked that theory out, its results always turned out in the end to agree with observation. It even let mathematicians predict results that the most eager imagination could hardly have guessed, and that experimenting at random might never have revealed. Dalton's laws of equivalent proportions in chemistry, if not his atomic theory, were founded on experiments made with the simplest and roughest apparatus, yet results deduced from them are verified every day in the most delicate processes of modern chemical analysis. The still more recent theory of the Conservation of Energy had been vaguely anticipated by Bacon, Rumford, Montgolfier, Seguin, Mayer and possibly others. Mr. Joule brought it to the test of experimental verification in some of the most beautiful and decisive experiments on record. It will be a long time before scientists have traced out all the consequences of this grand principle, but the way it agrees with the facts already puts it far beyond doubt.

I think it will now be clear that, although observation and induction must always be the ground of all certain knowledge of nature, on their own they could never have produced the results of modern science. A person who merely collects and digests facts will seldom come to understand their laws. A person who builds a theory and is satisfied with his own deductions from it, as Descartes did, will only astonish the world with his misused genius. The best student of science is the one who, with a plentiful store of theories and fancies, has the greatest power of foreseeing their consequences, the greatest care in comparing them with facts beyond doubt, and the greatest honesty in admitting the ninety-nine mistakes he has made on the way to the one true law of nature.

Questions and Exercises

1. Define an empirical law, and find a few more examples of such laws.

2. What are the three steps of the Deductive Method?

3. Trace some of the successive steps in the progress of the theory of gravitation, showing that it was established by this method.

Lesson 31: Explanation, Tendency, Hypothesis, Theory, and Fact

In the last few lessons I have used several expressions without ever defining what they mean. Now is a good time to show how these terms are used, with examples, and to get as clear an understanding as we can of their proper meanings.

Explanation literally means making something plain, or level, so that there is nothing uneven or obscure to block our view. Scientific explanation consists in harmonizing fact with fact, or fact with law, or law with law, so that we can see both to be cases of one uniform law of causation. Suppose we hear of a great earthquake in some part of the world, and later hear that a nearby volcano has erupted. We say that the earthquake is now partly explained. The eruption shows that great forces were at work beneath the earth's surface, and the earthquake is obviously an effect of causes like these. Or take the scratches that can be plainly seen on the surface of rocks in certain parts of Wales and Cumberland (in the English Lake District). They are explained by the fact that glaciers once existed in those mountains: the scratches match exactly the effects of the glaciers that still exist in Switzerland, Greenland and elsewhere. We may call these explanations of fact by fact.

A fact may also be explained by a general law of nature. That is, we can point out the cause and the way the fact was produced, and show that it is the same cause that operates in many cases that look quite different. So the cracking of glass by heat was explained (in Lesson 30) as one result of the universal law that heat increases the dimensions of solid bodies. The trade-winds are explained as one case of the general tendency of warm air to rise and be pushed aside by cold, dense air. The very same simple laws of heat and mechanics that make a draft flow up a chimney when there is a fire below also make the winds blow from each hemisphere toward the equator. At the same time, the simplest laws of motion explain why the winds come from the east. The earth rotates from west to east, and its surface moves much faster at the equator than nearer the poles. The air coming from the north or south tends to keep its slower rate of motion, so as the earth near the equator moves on beneath it, the wind seems to blow from east to west.

According to Mr. J. S. Mill, there are three distinct ways in which one law may be explained by other laws, or brought into harmony with them.

The first is the case where two or more separate causes are really at work, and their results are combined, or added together, homogeneously. As was explained before (in Lesson 29), the homogeneous intermixture of effects means that the joint effect is simply the sum of the separate effects, and is of the same kind as they are. Our last example, the trade-winds, really comes under this case. We find one law, or tendency, that makes winds blow from the arctic regions toward the equator, and a second tendency that makes them blow from east to west. These tendencies combine, and make the trade-winds blow from the northeast in the northern hemisphere and from the southeast in the southern hemisphere. Another example is the law that governs the temperature of the air in any part of the earth. It is a very complicated one. It depends partly on the law that governs the sun's heating power, and partly on the earth's power to radiate the heat away into space, but perhaps even more on the way currents of air or water bring warmth in or carry it away. The path of a cannon-ball, or any other projectile, is likewise fixed by several laws acting together. First comes the simple law of motion, by which any moving body tends to keep moving onward at a uniform rate in a straight line. Second comes the law of gravity, which keeps bending the body's path toward the earth's surface. Third comes the resistance of the air, which tends to reduce its velocity.

The reader may have noticed how often I use the word tendency, and that I have repeatedly spoken of a cause as tending to produce its effect. If the joint, homogeneous action of causes has been made clear, it will now be clear what a tendency is. A tendency means a cause that will produce an effect unless there are opposite causes which, combined with it, counteract and disguise that effect. When we throw a stone into the air, for instance, the attractive power of the earth tends to make it fall, but the upward motion we have given it disguises the result for a certain time. The endless circling motion of the moon around the earth is the result of two balanced tendencies: one toward the earth, and one to go straight on in a line. The laws of motion and gravity are such that this balance must always be kept. If anything brought the moon nearer to the earth, its tendency to fly off would increase, and would outweigh the effect of gravity until the moon had got back to its proper distance. A tendency, then, is a cause that may or may not be counteracted.

In the second case of explanation, an effect is shown to be due, not directly to the supposed cause, but to an intermediate effect of that cause. Instead of A being the cause of C, we find that A is the cause of B, and B the cause of C, so that B forms an intermediate link. This kind of explanation may seem to make the matter more complicated, but it really simplifies it. The connection of A with B may be a case of some familiar and simple law, and so may the connection of B with C, while the law that A produces C may be purely empirical and seem out of harmony with everything else. In lightning, for example, it looks as if electricity had the power of creating a loud explosion. In reality electricity only produces heat, and it is the heat that makes the sound, by suddenly expanding the air. So thunder comes into harmony with the sound of artillery, which is also caused by the sudden expansion of the heated gases given off by the gunpowder. Again, when chlorine was discovered, it was soon found to have a strong power of bleaching, and today almost all bleaching is done with chlorine instead of by the sun, as it used to be. Inquiry showed, however, that it was not really the chlorine that destroyed the color. Oxygen is the intermediate and active agent. Chlorine breaks down water, takes the hydrogen, and leaves the oxygen in a state of great activity, ready to destroy the organic coloring matter. In this way a number of facts are harmonized. We learn why dry chlorine does not bleach, and why several other substances resemble chlorine in their bleaching power: for instance ozone, hydrogen peroxide, sulfurous acid, and a peculiar oxide of vanadium recently discovered by Dr. Roscoe. We could not understand the effect at all unless we knew that in all these cases it is probably due to active oxygen, or ozone, even in the old method of bleaching by laying things out in the sun.

Footnote: Watts's Dictionary of Chemistry, Vol. i. p. 601.

The third case of explanation, and much the most important, is where one law is shown to be a case of a more general law. As was explained in Lesson 24, we naturally discover the less general laws first, and only gradually make our way to the simpler but deeper secrets of nature. It has often happened that scientists possessed several well-known laws without seeing the bond that connected them. People had long known, for instance, that all heavy bodies tend to fall toward the earth, and before Newton's time Hooke, Huygens and others knew that some force probably connected the earth with the sun and moon. It was Newton, however, who clearly brought these and many other facts under one general law, so that each fact, or less general law, throws light on every other.

The science of Electricity now harmonizes a vast series of partial laws and facts, among which it was truly hard to find any resemblance. The chief properties of the magnet had been fairly well known since the time of Gilbert, the physician of Queen Elizabeth. Ordinary frictional electricity was carefully studied by Otto von Guericke, Aepinus, Coulomb and others. Galvanism (electricity produced by chemical action, as in a battery) was investigated in great detail almost as soon as Galvani and Volta discovered that the chemical action of one substance on another can produce electricity. In the early part of this century there were three distinct sciences, Magnetism, Electricity and Galvanism; now there is only one. Oersted of Copenhagen supplied the first link between them in 1819, when he pointed out that an electric current can make a compass-needle move. Ampère and Faraday worked out the complicated relations among the three sciences, and in the end took them all into a wider science. We may call it Electro-magnetism, or perhaps, more conveniently, we may widen the name Electricity to take in all the phenomena connected with it.

A number of minor laws and separate facts are taken in and explained by the theory now generally accepted: that heat, electricity, light, and in fact all the phenomena of nature, are only different forms in which one and the same kind of energy shows itself. The total amount of energy in the universe is held to be fixed and unchangeable, like the quantity of matter. Sometimes it is disguised, because it affects only the molecules too small to perceive. At other times we see it produce visible mechanical effects, as in the fall of a stone or the expansion of steam. Now it had been known ever since the time of the Greeks that a simple lever greatly alters the character of a force, making its action slower or faster, but does not alter its amount: the more intense the force, the slower and more limited its action. In modern times a similar truth was proved of every kind of machine, and it was recognized that, apart from friction, no machine of any kind either creates or destroys energy. Quite separately, it had been recognized that the electricity produced in a galvanic battery is exactly proportional to the amount of chemical action, and that almost any one of the forces just named can be converted into any of the others. All facts like these are now taken in under one general theory. Its details are gradually being made more certain and accurate, but its main principle is that a certain amount of mechanical energy is equal to a certain amount of heat, or of electricity, or of chemical action, or even of muscular effort.

The word hypothesis is used a great deal in connection with the subject we are discussing, and we must consider what it means. It comes from the Greek words ὑπό (hupo), under, and θέσις (thesis), placing, so it means exactly the same as the Latin word suppositio, a placing under, which gives us our everyday word supposition. In science it seems to mean imagining some thing, force or cause that lies underneath the phenomena we are examining and is the agent that produces them, but that cannot be directly observed. When we make a hypothesis, we assert that a cause exists on the strength of the effects we observe. How probable it is that the cause exists depends on how many different facts or partial laws it lets us explain, or bring into harmony. To be of any value at all, a hypothesis must harmonize at least two different facts. Suppose we account for the effects of opium by saying, with Molière, that it possesses a dormitive power (a power of putting people to sleep), or say that a magnet attracts because it has a magnetic power. Everyone can see that we gain nothing. We know neither more nor less about the dormitive power or the magnetic power than we did about the opium or the magnet. But suppose we say that a magnet attracts because currents of electricity are circulating inside it. The hypothesis may seem very improbable, but it is valid, because it draws a certain analogy between a magnet and a coil of wire carrying electricity. Such a coil of wire attracts other coils in exactly the way one magnet attracts another, so this hypothesis lets us harmonize several different facts.

Take another example: the existence of intense heat in the interior of the earth. It is hypothetical in the sense that we cannot actually see the heat and measure it directly. But it harmonizes so many facts drawn from different sources that we can hardly doubt it is there. Hot springs and volcanoes are facts in its favor, though they might be explained on other grounds. The empirical law that the heat increases as we sink mines, in any part of the earth's surface, is stronger evidence. The intensely heated condition of the sun and the other stars strongly confirms it, because it shows that other bodies really do exist in the condition we suppose the earth's interior to be in. The cool state of the earth's surface is perfectly consistent with the earth's comparatively small size and with the known facts and laws about how heat is conducted and radiated. And the more we learn about the way the sun's heat is kept up by the fall of meteoric matter into it (the explanation most favored in the 1870s), the more probable it becomes that the earth, too, was once intensely heated like the sun, although it has been slowly cooling for an immense length of time. A supposition that agrees with so many facts, laws and other probable hypotheses almost stops being hypothetical, and its high probability makes us regard it as a known fact.

As long as it is consistent with the laws of thought, there is nothing we may not have to accept as a probable hypothesis, however hard it may be to imagine and understand. The force of gravity is hypothetical in the sense that we know it only by its effects on the motions of bodies. The way it weakens with distance does indeed match exactly the way that light, sound, electric or magnetic attraction, and in fact every influence that spreads out from a point through space, grows weaker. So it is probable that the law of the inverse square is absolutely true. But in other respects gravity goes strongly against all our ideas. If sound could travel to the sun as fast as it travels in the earth's atmosphere, it would take nearly fourteen years to get there. If the sun and the earth were joined by a solid, continuous bar of iron, a strong pull at one end would not be felt at the other for nearly three years. Light, it is true, comes from the sun in a little more than eight minutes. But what are we to think of the force of gravity, which seems to reach the sun in an instant, so short that no calculation has yet been able to detect any interval at all? In fact there seems to be some reason to suppose that gravity is felt instantaneously throughout the immeasurable regions of space.

The undulatory hypothesis of light (the hypothesis that light is a wave) has features just as extraordinary and inconceivable. That light really does consist of tiny but extremely rapid vibrations of something that fills space is almost certain. This hypothesis brings great harmony into the enormously varied and complicated phenomena of light, and it explains the analogy between light and sound. It is hard, of course, to imagine that anything can vibrate so fast as to strike the retina of the eye 831,479,000,000,000 times in one second, which is what must happen with violet light according to this hypothesis. But that is nothing compared with the difficulty of imagining space filled with a solid ether (the medium that was then believed to carry light), extremely rigid and elastic, which nevertheless offers no noticeable resistance to ordinary matter passing through it, and has no weight of its own. It has indeed been claimed that comets come back later than they should because of friction against this ether, and Mr. Balfour Stewart believes he has produced heat by the friction of a metal disc against the ether in a vacuum. If these claims prove true, we will have new facts in harmony with the theory of light, which will then be less hypothetical than before.

Footnote: See Sir John Herschel's Familiar Lectures p. 315, &c.

Now there is no difficulty in seeing the part that hypothesis plays in the deductive method of scientific investigation, which we considered in the last lesson. The preliminary induction is replaced, more or less completely, by imagining agents that we think could produce the known effects in question. Suppose our object is to explain what causes ebbing and flowing wells, which are found in many parts of the world (springs that flow and stop, or rise and fall, at intervals). We cannot possibly begin by exploring the inside of the earth until we find the source of a spring and observe its circumstances. We have to imagine cavities and channels of various shapes, until we think of an arrangement that would, according to the known laws of hydrostatics, make the water flow irregularly in the way we observe. If we can show that cavities of a particular shape will produce that effect, and can think of no other way it could be produced, the hypothesis becomes established as almost a certain fact.

It is the same with any great hypothesis, such as the theory of light. We have no way of directly observing and measuring the qualities of the ether that is the medium of light. All we know about this ether at present comes from the observed phenomena of light. So we are driven to invent something and give it qualities from which we can calculate, by certain principles of mechanics, what effects to expect. Then, finding that these effects can be made to harmonize with the effects actually observed, we rely on this agreement to prove that the ether exists. The truth of a hypothesis therefore depends entirely on its later verification, its agreement with observed facts. To invent hypotheses that cannot be verified in this way, or to invent them and then neglect to verify them, leads either to no result at all or to fallacy. But when the verification is careful and complete, no one can fairly object to the use of hypothesis. It may then be as certain as any other way of investigating, and it is indispensable in any case. In fact there was little truth or reason in Newton's famous protest against the use of hypothesis, "Hypotheses non fingo" (I do not frame hypotheses). The truth is that his theory of gravitation rested on the greatest and most successful of all hypotheses, while his views of the material nature of light, and of the causes of its peculiar phenomena, involved a false hypothesis that has long since been completely disproved.

The word theory has been used constantly in the last few lessons, and it deserves some examination. It comes from the Greek θεωρία (theoria), meaning contemplation, reflection or speculation, but that gives us little clue to how we use it now. In reality the word is highly ambiguous. Sometimes it is used as the equivalent of hypothesis, and at other times as the equivalent of a general law or truth. When people form theories about comets, the sun, the cause of earthquakes and so on, they imagine a great many things that may or may not exist. Theories like these are really complicated hypotheses, and should be called so. In this sense there are two theories of electricity. One supposes that there is a single fluid, which collects in some places and then tends to discharge itself toward places where there is too little of it, just as water always tends to find its level. The other supposes that there are two fluids, which are usually united but, when separated, tend to rush back together again. These so-called theories are really hypotheses, because we have no independent evidence that any such fluid exists, and it is now almost certain that there is no such thing. The atomic theory, again, is really a hypothesis put forward by Dalton to explain the remarkable laws he discovered in the proportions in which chemical elements combine. It is a valid hypothesis insofar as it really does explain why the quantities that combine are fixed. But as to the shapes, properties or absolute sizes of the atoms, it is purely hypothetical, because we have no facts in those respects for it to harmonize, and no apparent way of getting any.

In another, and more proper, sense, theory is opposed to practice, just as the general is opposed to the particular. The theory of gravitation means all the more general laws of motion and attraction on which Newton founded his system of the universe. We may know what those laws are without being able to work out the position of a planet or put them to any practical use. Skilled astronomers must calculate the particular results before navigators, travelers or anyone else can use them in practice to find latitude or longitude. When we speak of the mathematical theory of sound, the lunar theory or the theory of the tides, the word has no special reference to hypothesis. It simply means general knowledge, or science. It implies that we possess a complete series of general and accurate laws, but does nothing to set them apart from accurate knowledge in general. When a word like theory really is used in an equivocal way, it is not wise to try to give it an exact definition, which would be imaginary and artificial.

The word fact is used very often in this book, as in most books, and calls for a few remarks. It comes from factum, the past participle of facere, to do, and so would mean something done, an act or deed. But its meaning has obviously been greatly extended by analogy. We usually set fact and theory against each other. But just as theory seems to have two meanings that get confused, I believe that fact is ambiguous too. Sometimes it means what is certain and known by the evidence of the senses, as opposed to what is known only probably, by hypothesis and inference. At other times it is contrasted with a general law, and means a particular instance or case. A law of great generality may often be just as certain and true as the particular facts that come under it, especially in mathematics, so in this case the contrast must be between the general and the particular. We also often use the word in everyday life simply to mean a truth. We might say, for example, "It is a fact that the primary laws of thought are the foundation of reasoning." In short, just as theory means, ambiguously, what is hypothetical, general, abstract or uncertain, so fact is equally ambiguous, and means, in a confused way, what is intuitively known, particular, concrete or certain.

See Mill's System of Logic, Book III, Chapters 12, 13 and 14, Of Explanation, and Hypothesis.

Questions and Exercises

1. What do you mean by the explanation of a fact?

2. State the three ways in which a law of nature may be explained, and suggest some more examples of each case.

3. Define tendency. Do all causes consist only of tendencies, or can you find examples to the contrary?

4. Give a definition of hypothesis. How can a valid hypothesis be told apart from an invalid one?

5. What place does hypothesis hold in the Deductive Method?

6. Explain the ambiguities of the words theory and fact.

Lesson 32: Classification, and Abstraction

In Lesson 12, on the Predicables, we looked at the doctrine of classification as logicians treated it many centuries ago. Over the last two centuries, though, the progress of science has drawn a great deal of attention to the true principles by which we can put a huge number of different objects in order. So we now have to ask what makes a system of classification natural and perfect.

In fact, you could say that our subject here covers the whole science of logic. All thought and all reasoning, as far as it deals with general names or general notions, can be said to consist in classification. Every common, or general, name is the name of a class, and every name of a class is a common name. "Metal" is the name of one class of substances, the one we have used so often in our syllogisms; "element" is the name of another class, of which the first is a part. Reasoning has been described, quite plausibly, as affirming of the parts of a class whatever can be affirmed of the whole. Every law of nature we arrive at lets us class a number of facts together. It would hardly be going too far to define logic as the theory of classification.

Here, however, we are dealing with the more deliberate and clear-cut arrangement of objects or notions that is used above all in the natural sciences, such as botany, zoology, mineralogy and paleontology.

The history of the word class is rather curious. In ancient Rome the whole people was summoned together at certain times, and this ceremony was called a classis, from the Greek κλάσις (klasis) or κλῆσις (klēsis), which comes from καλέω (kaleō), to call together. Servius Tullius is said to have divided the people into six orders according to how much tax they could pay, and these orders were, naturally enough, called the classes of the people. From there the name gradually came to be used for any organized body of people, such as an army. Then it was carried over to a fleet of ships drawn up in a fixed order, and finally, by analogy, it was stretched to cover any collection of objects carefully arranged. It is curious that when we speak today of the lower or higher classes of the people, we are bringing the word back very close to its original meaning.

Classification is perhaps best defined as the arrangement of things, or of our notions of them, according to their resemblances or identities. Every class should be made up of objects that resemble each other exactly in certain definite qualities, and those qualities are stated in the definition of the class. The more numerous and wide-reaching the resemblances that a system of classes points out, the more perfect and useful we must consider that system.

Mr. J. S. Mill describes his view of what classification means this way: "Classification is a contrivance for the best possible ordering of the ideas of objects in our minds; for causing the ideas to accompany or succeed one another in such a way as shall give us the greatest command over our knowledge already acquired, and lead most directly to the acquisition of more. The general problem of classification, in reference to these purposes, may be stated as follows: To provide that things shall be thought of in such groups, and those groups in such an order, as will best conduce to the remembrance, and to the ascertainment of their laws."

A collection of objects can usually be classified in an unlimited number of ways. Any quality that some of them have and others lack can be taken as the first difference, and the groups it marks off can then be subdivided, one after another, by any other qualities we like. A library of books, for example, could be arranged (1) by size, (2) by the language they are written in, (3) by the alphabetical order of their authors' names, (4) by subject, and in various other ways. Large libraries and catalogues use these arrangements and combine them in different ways. Each arrangement has its own particular convenience, and we must choose the one that best serves the special purpose of the library or catalogue. The population of a country, again, can be classified in an almost endless number of ways, for different purposes or different sciences. The people of the United Kingdom can be divided by place of birth, as English, Welsh, Scottish, Irish, colonial-born, and foreigners. The ethnographer would divide them into Anglo-Saxons, Cymri (the Welsh Britons), Gaels, Picts, Scandinavians, and so on. The statistician arranges them by age; by marital condition, as married, unmarried, widowed, and so on; and by state of body, as able-bodied, incapacitated, blind, or intellectually disabled. The economist looks at the countless trades people carry on and classifies them in a complicated way. The lawyer, again, treats everyone as a minor, an adult, a feme sole (an unmarried woman, who held property in her own name), a feme couverte (a married woman, whose legal identity the law then merged with her husband's), a guardian, ward, trustee, felon, and so on.

In the natural world, too, we can make various classifications. Plants can be arranged by the country they come from; by the kind of place, or habitat, they thrive in; by how long they live, as annual, biennial, perennial; by size, as herbs, shrubs, trees; or by their uses, as food plants, drugs, or poisons. All of these are different from the classifications that botanists devise to represent the natural affinities, or relationships, of plants. So it is clear that when we make a classification there is no single fixed method that can be found by rule. Usually an unlimited number of choices are open to us. In cases like this logic cannot do much; it is really the job of the special sciences to work out what kind of classification is needed. All logic can do is point out certain general requirements and principles.

The first requirement of a good classification is that it suits the purpose in hand. That is, the points of resemblance chosen to form the leading classes should be the ones that matter for the practical use of the classification. Everything that needs to be treated alike must be grouped together, and things that need to be treated differently must be separated. A lawyer, for instance, has no need to classify people by the English county they were born in, because the law is the same whatever the county. But a Scot, a Manxman, or a foreigner lives under laws different from those of a man born in England, and to that extent we will have to classify them separately. A gardener is quite right to classify plants as annuals, biennials, perennials; as herbs, shrubs, trees; as evergreen and deciduous; or by the soil, temperature and other conditions that affect them. These are exactly the points that must guide him in treating some plants differently from others.

A second requirement, and from a scientific point of view the most important one, is that a good classification lets us make the greatest possible number of general assertions. This is the criterion, as Dr. Whewell stated it, that distinguishes a natural system of classification from an artificial one, and we need to look carefully at what it means. Clearly, a good classification is more than just an orderly arrangement. It involves a process of induction that brings to light all the more general relations among the things being classified. An arrangement of books will usually be artificial: the octavo volumes have nothing in common except being octavo size. An alphabetical arrangement of names, again, is extremely convenient and suitable for many purposes, but it is artificial, because it allows few or no general assertions. We cannot make any general assertion at all about people just because their names happen to begin with an A or a B, a P or a W. Even the people who share the name Smith or Taylor or Robinson could be put through the inductive method of agreement without our discovering any common circumstance that could be stated as a general proposition, or law. It is true that if we looked into the backgrounds of the Evanses and the Joneses, we would find nearly all of them to be Welsh, and the Campbells to be Scottish, and people with a very unusual name would often turn out to descend from common ancestors. To that extent even an alphabetical arrangement has something natural in it, and lets us make general assertions. In fact, hardly any arrangement can be made that does not show some traces of important relations and resemblances. But what we want is a system that will reveal all the most important general truths.

For this purpose we must choose, as the basis for grouping things together, the characters that carry the most other characters with them. In Lesson 12 we described a property (proprium) as a quality that belongs to the whole of a class without being part of the class's definition. Now, we should frame the definition of a class so that it contains as few characters as possible, while as many other characters, properties, or propria as possible can be attributed to the things in the class. Everyone can see, for instance, that animals form one great group of living beings with many characters in common, and that plants form another. Animals have sensation and voluntary motion; they consume carbon-containing food and give off carbonic acid (carbon dioxide); they have a stomach; and they produce fat. Plants lack sensation and voluntary motion; they build carbon-containing tissue, absorb carbonic acid, and give off oxygen; they have no stomach; and they produce starch. At one time it might have been thought that almost any one of these characters was enough to tell which group a living being belonged to. Whatever had a stomach was an animal, and whatever did not was a plant. Whatever produced starch or gave off oxygen was called a plant; whatever absorbed oxygen or produced fat was an animal. These statements are still generally true today, so we can make assertions in the form of the proposition U (the doubly universal affirmative of Lesson 22): "all animals are all beings that give off carbonic acid, and all plants are all beings that absorb carbonic acid." But in reality there are many exceptions, and the more research we do, the clearer it becomes that no definite line can be drawn between animal and vegetable life. This, of course, is not a failure of logical science. It is a highly significant fact about the things themselves.

When we classify plants, we again meet very deep and natural distinctions, this time between the great classes called Exogens, Endogens, and Acrogens (roughly, the groups botanists now call dicots, monocots, and the spore-bearing plants such as ferns and mosses). The Acrogens have no true flowers with sexes and no seeds. They are made almost entirely of cellular tissue, and their outer skin, or epidermis, has no pores. The other two classes have much in common: they have true flowers, woody tissue and pores in the skin, and so they can be united into one wider class, the Vasculares. But exogens and endogens are also very sharply distinguished from each other. Exogens have a stem or trunk made of distinct bark, pith, and wood in concentric layers; leaves with net-like veins; and seeds with two seed-leaves and a naked radicle (the root of the embryo). Generally, too, the parts of the flower come in some multiple of two or five. Endogens, on the other hand, have no distinct bark, pith, and wood, and no concentric layers; their leaves have parallel veins; their seeds have one seed-leaf and a radicle that is not naked; and the parts of the flower generally come in multiples of three.

These are the very widest classes in what is called the natural system of botanical arrangement, but the same principles hold in all its smaller classes. Botanists are constantly working to bring the great multitudes of plants together into species, genera, orders, classes, and various groups in between, so that the members of each group have the greatest number of points of resemblance to each other and the fewest points of resemblance to members of other groups. This is how the great purpose of classification is best fulfilled. Classification reduces multiplicity to unity, and it lets us infer of all the other members of a class what we know of any one member, provided we distinguish properly between the qualities that are likely, or known, to belong to the class and those that are peculiar to the individual. As Professor Huxley has remarked, it is a necessary condition of correct classification that the definition of a group should hold exactly true of all the members of the group, and not of the members of any other group. In the natural sciences, though, this condition is very hard to carry out, because kinds of plants or animals keep being discovered that sit halfway between classes that would otherwise be well distinguished. Ferns, for example, are a real embarrassment to the fundamental division of plants. They have no true flowers, and in this and other ways they agree with the other acrogens; yet they have plenty of woody fiber, which would entitle them to rank with the vasculares, the larger group that exogens and endogens are subdivisions of.

We might add that the progress of chemistry is quickly turning it into a science of classification. In fact the whole theory of chemical combination now depends on grouping the elements and compounds correctly. Dr. Roscoe, in his Lessons in Elementary Chemistry, lists no fewer than eleven classes of metals, each class having a number of properties in common. The metals of the alkalies, for instance, namely potassium, sodium, cesium, rubidium and lithium, form a remarkably natural class. They are all soft, melt easily, and turn to vapor at high temperatures. They combine very violently with oxygen and decompose water at any temperature, forming oxides that dissolve very readily in water and become powerfully caustic and alkaline substances, from which heat cannot drive the water out. Their carbonates dissolve in water, and each metal forms only one compound with chlorine.

The metals of the alkaline earths, calcium, strontium and barium, also form a very natural class. What distinguishes them is that their carbonates do not dissolve in pure water but do dissolve in water that has carbonic acid dissolved in it. The gold class contains the rare or valuable metals gold, platinum, palladium, rhodium, ruthenium, iridium and osmium. Nitric acid does not act on them, and they can only be dissolved by chlorine or by the mixture of acids called aqua regia. Their oxides can be reduced, or have their oxygen removed, simply by heating them.

Natural classifications give us the deepest resemblances and relations, and they may lead us in the end to know how the different varieties of things came to be produced. So they are essential to a true science; you could almost say they make up the framework of the science. It does not follow, though, that they suit every purpose. When all we want is to recognize which chemical element, plant or animal we have in front of us, its character as defined in a natural system will give us little or no help. A chemist does not detect potassium by getting it into the state of a metal and testing whether it decomposes water. He simply looks for whichever compounds of potassium have the most clearly marked and unusual characters. A compound of potassium, platinum and chlorine, for example, is the most distinctive, or characteristic, of the metal, and it is generally used as a way of recognizing it. The fine violet color that potash (a potassium compound) gives to the flame of a lamp was also used as a sign of potassium's presence long before the spectroscope was brought in to analyze such colors. An artificial classification of the elements is therefore necessary for detecting substances. Accordingly, any book on chemical analysis will contain arrangements of the elements by characters of very minor importance, which are chosen because they can be observed easily and reliably.

In botany, again, the natural system of classification is far from well suited to finding out the name of a plant. Its classes are often defined by the shape of tiny parts of the seed, the arrangement of the seed-vessel, and other parts that are usually hard, and sometimes impossible, to examine. So botanists usually arrange their genera and species in the order of the natural system, but devise a sort of key, or artificial arrangement, that uses the simplest and most obvious characters, often called characteristics, to tell the plants apart. The best arrangement of this kind for British plants is found in Bentham's British Flora. In fact, the famous Linnaean arrangement of plants was intended by its author to serve exactly this purpose. Linnaeus was far too deep a thinker to suppose that the numbers of stamens and pistils generally expressed the real relationships of plants. Many of his classes were genuinely natural classes, but he chose the stamens and pistils as the general guide to the classes and orders because they were very plain and obvious marks.

Closely connected with classification is the process of abstraction. To abstract is to separate the qualities common to all the individuals of a group from the peculiarities of each individual. The notion "triangle" is the result of abstraction, to the extent that we can reason about triangles without paying any attention to the particular size or shape of any one triangle. All classification involves abstraction, because in forming and defining a class I must separate the common qualities from the peculiarities. When I abstract, I also form a general conception, that is, one that usually takes in many objects. But if the quality I abstract is a peculiar property of the class, one that belongs to the whole class and to no other objects, I may not increase the extent of the notion at all. So Mr. Herbert Spencer is perhaps right to hold that we can abstract without generalizing. We often use this word generalization, and the process can be defined as inferring of a whole class what we know only of a part of it. Whenever we regard the qualities of a thing as not confined to that thing alone but as extending to other objects, and in fact whenever we consider a thing only as a member of a class, we are said to generalize. If, after studying the properties of the circle, we go on to those of the ellipse, the parabola and the hyperbola, we soon find that the circle is just one case of a whole class of curves called the conic sections, which correspond to equations of the second degree. I generalize when I regard certain properties of the circle as shared by many other curves.

Dr. Whewell added to the oversupply of terms for the same processes when he introduced the expression Colligation of facts. Whenever two things are found to have similar properties, so that they are placed in the same class, they can be said to be connected together. We connect together the positions of a planet as it moves around the sun when we think of them as points on one and the same ellipse. Whenever we join together facts that were unconnected before, by means of a suitable general notion or hypothesis, we are said to colligate them. Dr. Whewell adds that the general conceptions we use must be (1) clear and (2) appropriate. But it is very doubtful whether there is anything in these processes really different from the general process of natural classification that we have been considering.

Questions and Exercises

1. Define classification, and give the derivation of the word.

2. What do you mean by important characters in classification?

3. State Dr. Whewell's criterion of a good natural arrangement.

4. Distinguish between a natural and an artificial system of classification.

5. What do you mean by a characteristic quality? Is it always an important quality?

6. Define abstraction, generalization, and colligation of facts.

7. What are the characters of a notion that has been properly abstracted?

Lesson 33: Requisites of a Philosophical Language

Among the supporting processes that inductive reasoning needs if it is to succeed, we must count the building of a suitable language. In fact, it is impossible to overstate how much any science depends on a language that is both accurate and rich. Studying things would be almost useless if we had no names to mark those things and to record what we have observed about them.

It is easy to see that language serves three distinct and almost independent purposes:

1. As a means of communication.

2. As a mechanical aid to thought.

3. As an instrument of record and reference.

When language first arose, it was used mainly, if not only, for the first purpose. In the 1870s, many of the world's peoples were thought to build up no store of knowledge at all. We may even say that the lower animals often have some way of communicating by sounds or natural signs, and this amounts to language in the first sense, even though they cannot reason with general notions.

Some philosophers have held that reasoning is impossible without language. The thoroughgoing nominalist went so far as to say that there are no such things as general notions at all, so that general names are the only thing that is general in science and reasoning. That is no doubt false (see Lesson 2). Still, we have to admit that if our general ideas were not fixed and represented by words, we could never manage the kind of sustained thought we now enjoy. For practical purposes, the second use of language is certainly indispensable, and reasoning can almost be thought of as the same thing as the correct use of words. When language is used only to help us reason, there is no need for each word to have a fixed meaning. We could use names the way algebra uses the letters x, y, z, a, b, r and so on, to stand for whatever quantity happens to turn up in a problem. All we need is never to confuse the meaning we give a word in one argument with a different meaning we give it in another. Algebra, in fact, can be described as a very perfect kind of language built for the second purpose alone, one that can lead a person to the solution of a problem in a symbolic, mechanical way.

Language as it comes to us ready-made, grown by habit over the centuries, can serve all three purposes, though by no means perfectly. Because words have a more or less fixed customary meaning, we can not only reason with them but also communicate our thoughts or record them. It is this last use, the recording, that we now have to deal with.

Building a science takes far more facts than anyone could hold in memory with enough accuracy. So one of the indispensable aids to induction is a means of describing and recording our observations. Only in this way can knowledge pile up, so that each observer starts with the advantage of knowing what has already been recorded and proved. We need, then, to look at the way language serves to register facts, and to work out what qualities a philosophical language must have to meet the needs of science.

As an instrument of record, language clearly needs two main things:

	1. Precision or definiteness of meaning.
	2. Completeness.

A name is worse than useless if, when it is used to record a fact, it does not let us find out what kind of fact was recorded. So accuracy and precision matter more in a language than abundance. Lacking the right word will seldom lead to actual error and fallacy. It will only force us to use a roundabout phrase, or else leave the fact unrecorded. But it is obviously convenient that whenever a thing, notion or quality has to be mentioned often, there should be a name set aside for it, and only one name. Let us look in turn at what a precise and complete language must be like.

The reader may not have noticed it before, but it is certainly true: we cannot describe anything without asserting that it resembles some other thing. We can only describe a thing by giving it a name. But how are we to learn what that name means? If we explain the name with other names, we only end up with more names whose meanings we need. In the end we must learn meanings, not from names, but from the things that carry those names. If someone did not know what blue means, there would be no way to tell them except by pointing to something that gave them the sensation of blueness. If they had been blind from birth, they could never get any notion of what blueness is. It is true that many words are so familiar to us from childhood that we cannot say when or how we learned what they mean, though we must have learned it by reference to things. But as soon as we start using names more precisely, we quickly have to refer to physical objects again. So we describe the various kinds of blue as sky-blue, azure-blue, indigo-blue and cobalt-blue, and we likewise tell apart sea-green, olive-green, emerald-green, grass-green and so on. In botany, the shapes of leaves are described with words like ovate, lanceolate, linear, pinnate and peltate, which call to mind, in order, an egg, a lance, a line, a feather and a shield. When we record sizes, we equally cannot avoid comparing them with the sizes of other things. A yard or a foot means nothing unless there is a definite standard yard or foot somewhere that fixes its meaning, and the reader probably knows that once the physical standard of a length is completely lost, it can never be recovered. The word is nothing unless somewhere we have the thing it stands for.

So the first requisite of a philosophical language is plainly that "every general name must have a certain and knowable meaning." It hardly needs saying that singular or proper names, the names of particular objects, must be known too. But such names are just marks stuck on the things they name, so they do not need the same attention. General names are harder, because, as we saw in Lesson 5, they have a double meaning: their denotation, or extension, and their connotation, or intension. Of these two, it is the connotation that must be fixed. The denotation, as a general rule, cannot be limited and pinned down. Suppose the name planet had been restricted to Jupiter, Saturn, Mars, Venus and Mercury, the planets known before the telescope was invented. Then we would have needed a new name for the ones discovered later, and even so we would be making the mistake of giving different names to things that are very much alike. But if by planet we mean any round body that revolves around the sun in a slightly elliptical orbit, the name will take in every such body that is discovered from time to time. In the 1870s more than 100 were already known, counting the small planets we now call asteroids. In the same way, locomotive engine is not just the name of a certain number of engines that exist right now. If it were, we would need a new name every week, as each new engine was built or an old one scrapped. What is fixed in a general name is its connotation, the qualities implied in the things that bear the name. So as far as possible we ought to define every general name we use, not by naming the objects it denotes, but by naming the qualities it connotes. We have already dealt with definition in earlier lessons (Lessons 12 and 13), so here we only need to ask how far it is better to use words already in everyday use rather than newly invented terms.

The advantage of an old term is that it carries meaning for everyone, and to that extent it saves people from having to learn a strange technical expression. Everyone knows what heat is, and the phrase "the science of heat" means something to every person, however unlearned. But old terms have one drawback we must note: they are almost always open to ambiguity. And in fact the scientist uses the word heat differently from everyone else. In science, all things are more or less hot, whereas in everyday life we could never say that ice was hot or contained heat. Ordinarily, heat means a temperature above the usual average, and the idea only makes sense in contrast with cold. We also stretch the word by analogy to sensations of taste, as when we say pepper is hot, and even to things purely in the mind, as in a hot dispute, a hot temper and so on. If we invent a new term, Caloric, to avoid these ambiguities, we can give it as precise a meaning as we like. But we put one more obstacle in the way of studying science, because there is one more technical term to learn.

This difficulty is especially great in the science of political economy (what we now call economics). There we deal with such familiar ideas as wealth, money, value, currency, capital, labor and exchange. It is the very familiarity of these ideas that causes the greatest trouble, because different people attach different meanings to the words, and the result is endless logomachy (from the Greek λόγος (logos), word, and μάχη (machē), battle), that is, disputes that arise over merely verbal questions. Even when a writer carefully defines the sense in which he uses these terms, he cannot make other people keep his definitions in mind. The other option, inventing entirely new terms, is out of the question, since it would certainly make a book unbearable for most readers. The only advice that can be given is this: bring in a new term where it is likely to be readily accepted and to push out an old, ambiguous one; otherwise, try to remove the ambiguity of the old term by always keeping a precise definition of the intended meaning in view.

A complete philosophical language will be made up of two distinct kinds of terms, which form, respectively, the descriptive terminology and the nomenclature of the science.

A descriptive terminology, as Dr. Whewell pointed out, must include all the terms needed to describe exactly what has been observed about any object or phenomenon, so that we have a permanent record of the observation. For every quality, shape, circumstance, degree or quantity there must be a suitable name or way of expressing it. So when we record the discovery of a new mineral, we ought to be able to pin down in words its exact crystal form, its color, its degree of hardness, its specific gravity, its smell and taste if it has any, and many other qualities that may turn out to matter. Modern botany grew out of the efforts of Linnaeus to create a system of terms with which every part and feature of a plant can be described accurately. The language of botany, improved since his time, is the most complete example of a scientific terminology. Geology, in my view, suffers badly from the difficulty of finding accurate terms. Names such as trap, basalt, gneiss, granite, tuff, greenstone, trachyte, porphyry, lava and so on are very vague, and there are no precise descriptive terms to define them and tell them apart. Where a quality does not come in degrees or quantities, it needs only a single name. Otherwise we must find some way of measuring it exactly and expressing the measurement. Inventing an instrument to measure a quality that was never measured before is always an important step in science, and the making of the thermometer by Fahrenheit and of the pendulum clock by Huygens were turning points in science.

On the other hand, each science needs a nomenclature, a collection of names for the distinct objects, or classes of objects, it deals with. In mineralogy the names of the separate minerals, such as haematite, topaz, amphibole, epidote, blende and polybasite, make up the nomenclature. In chemistry we have all the names of the elements, together with a vast apparatus of names for organic and other compounds, such as ethyl, acetyl, cyanogen, naphthalin, benzol and so on. In astronomy the names of the planets, satellites, nebulas, constellations and individual stars form a nomenclature, though by no means a perfect or convenient one. Geology likewise has a nomenclature that is necessarily incomplete, in the names of its successive formations: silurian, devonian, carboniferous, permian, triassic, eocene, miocene, pliocene, post-pliocene and so on.

Clearly a nomenclature must have names at various levels of generality. It must include individual objects, if they need to be recorded separately; the infimæ species, if there are any; and wider classes above them, all the way up to the summa genera, the widest notions the science covers. In astronomy we deal mostly with the names of individual objects, and so far there is little room for classification. In natural sciences such as botany or zoology there is seldom or never any need for names of individuals. That is because, in these sciences, a countless multitude of individuals generally resemble one another very closely in a great many properties, so that they make up what has been called a natural kind. Mr. J. S. Mill uses this term to mean "one of those classes which are distinguished from all others, not by one or a few definite properties, but by an unknown multitude of them; the combination of properties on which the class is grounded being a mere index to an indefinite number of other distinctive attributes."

Going by the way Mr. Mill puts it, he seems to include in a nomenclature only the names of supposed species. For he says: "A nomenclature may be defined, the collection of names of all kinds with which any branch of knowledge is conversant; or more properly, of all the lowest kinds, or infimæ species, those which may be subdivided indeed, but not into kinds, and which generally accord with what in natural history are termed simply species." But the fact is that naturalists have now given up the idea that a species is any fixed form. Many species are already divided into subspecies and varieties, or even varieties of varieties, and on the principles of Darwin's theory the dividing could go on without end. Surely the most reasonable view is that the natural kingdoms of plants and animals are arranged in an endless series of classes and subclasses, and that all the names attached to any of these classes belong to the nomenclature.

Again, Mill does not count as part of the nomenclature those general names that stand for ideas formed artificially in the course of induction and investigation. So, besides a terminology fit for describing the individual facts we observe with precision, there is, in his account, a branch of language containing "a name for every common property of any importance or interest, which we detect by comparing those facts: including (as the concretes corresponding to those abstract terms) names for the classes which we artificially construct in virtue of those properties, or as many of them, at least, as we have frequent occasion to predicate any thing of." As examples of this kind of name he gives Circle, Limit, Momentum, Civilization, Delegation and Representation. While the nomenclature contains the names of natural classes, this third branch of language would apparently contain the names of artificial ideas or classes.

But I find it very hard to give a clear account of Mill's views on this subject. And since my aim in these lessons does not allow me to discuss unsettled questions, I must end by referring any reader who wants to take the subject further to the 4th and 6th chapters of the 4th Book of Mr. Mill's System of Logic, which deal with the Requisites of a Philosophical Language.

See Dr. Whewell's "Aphorisms concerning the Language of Science," at the end of his Philosophy of the Inductive Sciences.

Thomson's Outline of the Laws of Thought contains very interesting remarks on the general nature and use of language, §§ 17–31.

Questions and Exercises

1. What are the three purposes for which we use language?

2. What are the two chief requisites of a philosophical language?

3. What should guide us when we choose between a new scientific term and an old one?

4. Explain the difference between a Descriptive Terminology and a Nomenclature. Then sort the following terms according to which of the two they belong to:

Rose, Rosaceae, Rose-like, Potassium, Alkaloid, Ruminant Animal, Ruminating, Ruby, Ruby-red.

5. What does Mill mean by the expression Natural Kind?

A Vocabulary of Logical Terms

Abscissio Infiniti (the cutting off of the infinite or negative part), the process by which we find the place of an object in a system of classes, by comparing it with one class after another and rejecting each class it does not belong to.

Accidental definition is a definition that gives the properties of a species, or the accidents of an individual; it is more commonly called a Description.

Analogue, a thing that is analogous to some other thing.

Analytics (τὰ Ἀναλυτικά), the title given in the second century to parts of the Organon, the logical treatises of Aristotle; they were divided into the Prior and the Posterior Analytics.

Analytic syllogism, a syllogism in which the conclusion is put first and the premises follow as the reasons for it. See Synthetic syllogism; the distinction is unimportant.

Antinomy (ἀντί, against; νόμος, law), the opposition of one law or rule to another. Kant.

Argument (Latin, argus, from ἀργός, clear, manifest), the process of reasoning: showing or proving what is doubtful by means of what is known. See Inference (Lesson 10). The middle term of a syllogism is sometimes called the argument in a special sense.

Argumentum a fortiori, an argument in which we prove that the case in question is even stronger or more probable than one already admitted to be strong or probable enough.

Argumentum ad judicium, an appeal to the common sense of humankind.

Argumentum ad ignorantiam, an argument that rests on the ignorance of one's opponents.

Argumentum ad verecundiam, an appeal to our respect for some great authority.

Argumentum ex concesso, a proof drawn from a proposition the other side has already granted.

Assertion (ad, to; sero, to join), a statement or proposition, whether affirmative or negative.

Association of ideas (associo, to accompany; socius, a companion), the natural connection in the mind between impressions that have been together before, or that are similar. Any idea tends to bring its associated ideas into the mind, following the two great laws of association: the Law of Contiguity and the Law of Similarity.

Assumption (assumo, to take for granted), any proposition taken as the basis of an argument; in a special sense, the minor premise of a categorical syllogism.

Attribute (attribuo, to give or ascribe to), a quality or circumstance that can be affirmed (or denied) of a thing; the opposite of Substance, which see.

Belief, assent to a proposition. It can have any degree of strength, from the slightest probability to the fullest certainty; see Probability.

Categories, the summa genera, the widest classes into which things can be sorted. There are ten of them: Οὐσία, Substance; Ποσόν, Quantity; Ποιόν, Quality; Πρός τι, Relation; Ποιεῖν, Action; Πάσχειν, Passion, or being acted on; Ποῦ, Place; Πότε, Time; Κεῖσθαι, Position; Ἔχειν, Habit or condition. Everything that can be affirmed must fall under one or another of these highest predicates. Aristotle described them in the first treatise of his Organon, called the Categories.

Cause, Aristotle distinguished four kinds of cause for the existence of a thing: 1. The Material Cause, the substance or matter it is made of; 2. The Formal Cause, the pattern, type or design it is shaped by; 3. The Efficient Cause, the force used in shaping it; 4. The Final Cause, the end, motive or purpose of the work.

Chance, ignorance of the causes that are at work; see Probability.

Cognition (cognosco, to know), knowledge, or the action of the mind in acquiring knowledge.

Colligation of Facts, Dr. Whewell's expression for binding facts together in the mind by some suitable conception.

Comparison (com, together; par, equal or like), the action of the mind by which we judge whether two objects of thought are the same or different in certain respects. See Judgment (Lesson 2).

Compatible terms are terms that are distinct but not contradictory, and so can be affirmed of the same subject, such as "large" and "heavy," or "bright-colored" and "nauseous."

Composition of Causes, the principle shown in every case where the joint effect of several causes is identical with the sum of their separate effects. J. S. Mill.

Concept, that which is conceived, the result of the act of conception; nearly the same as general notion, idea, thought.

Conception (con, together; capio, to take). An ambiguous term. Properly it means the action of the mind in which it takes several things together so as to form a general notion; or else the action in which it forms "a mental image of the several attributes given in any word or combination of words." Mansel.

Conjugate words, words that come from the same root or stock, such as known, knowing, knowingly, knowledge.

Consciousness, the immediate knowledge the mind has of its sensations and thoughts, and, in general, of everything it is doing at present. Reid.

Consectary = Corollary.

Consequence, the connection between antecedent and consequent; but often used, ambiguously, for the consequent itself.

Consilience of Inductions, the agreement of inductions drawn from different and independent series of facts, as when we learn that the earth moves by entirely different kinds of observation and reasoning. Whewell.

Contingent (contingo, to touch), that which may or may not happen; the opposite of both the necessary and the impossible.

Continuity, Law of, the principle that nothing can pass from one extreme to another without passing through all the degrees in between; motion, for instance, cannot be produced or destroyed in an instant.

Corollary, a proposition that follows directly from another that has been proved.

Criterion (κριτήριον, from κρίνω, to judge), any fact, rule, knowledge or means needed for forming a judgment that will settle a doubtful question.

Data (plural of datum, that which is given), the facts or assertions from which an inference is to be drawn.

De facto, what actually or really happens: the opposite of de jure, what ought to happen by law or right.

Demonstration (demonstro, to point out), strictly, pointing out the connection between premises and conclusion. The term is more generally used for any argument or reasoning regarded as proving the conclusion it asserts. A demonstration is either Direct or Indirect. In an indirect demonstration we prove the conclusion by disproving its contradictory, that is, by showing that the conclusion cannot be supposed untrue.

Determination, the marking off of parts of a genus by joining the genus and a difference together again. See Division (Lesson 12).

Dialectic (διαλεκτικὴ τέχνη, the art of discourse, from διαλέγεσθαι, to discourse). The original name of Logic, perhaps invented by Plato. It has also been used to mean the Logic of Probable Matter (Aristotle), the right use of Reason and Language, and the Science of Being, so it is a highly ambiguous term.

Disbelief, the state of mind in which we are fully convinced that some opinion is not true. J. S. Mill. It amounts to belief in the contradictory opinion or assertion, and must not be confused with Doubt, which see.

Doubt (dubito, to go two ways), the state of mind in which we hesitate between two or more inconsistent opinions. See Disbelief.

Drift of a proposition, the different meanings the same sentence can be given depending on where the stress falls. See Fallacy of accent (Lesson 20).

Empiricism (ἐμπειρία, experience), the doctrine of those who hold that all knowledge comes merely from experience.

Essence (essentia, from esse, to be), "the very being of anything, whereby it is what it is." Locke. It is an old scholastic word that cannot really be defined, and it should be dropped from use.

Evidence (e, and videre, to see), literally the seeing of anything. The word now means any facts taken in by the mind and made the grounds of knowledge and belief.

Experimentum crucis, an experiment that decides between two rival theories and shows which one to adopt, just as a signpost (a "finger-post") shows which of two roads to take.

Exposita, a proposition given to be worked on by some logical process.

Extremes of a proposition are its ends or terms, the subject and the predicate.

Fundamentum relationis, the ground of a relation, that is, the series of events or circumstances that sets up a relation between two correlative terms.

Galenian or 4th figure of the syllogism; see Lesson 17.

Homologue, whatever is homologous.

Homology, a special term for the analogy between parts of different plants and animals, such as between the wing of a bird and the foreleg of a four-footed animal, or between the scales of a fish and the feathers of a bird.

Idea (ἰδέα, εἶδος, image), a term used ambiguously, but usually equivalent to thought, notion, concept. Locke defines it as "Phantasm, notion, species, or whatever it is which the mind can be employed about in thinking." To have an idea of a thing is to think of that thing.

Idol (εἴδωλον, εἶδος, image), Bacon's figurative name for the sources of error. He listed four kinds: Idols of the Tribe, which affect all people; Idols of the Cave, which belong to one individual; Idols of the Forum, which arise from people's dealings with one another; and Idols of the Theatre, which come from the systems of philosophers.

Illation (illatum, past participle of infero, to bring in). See Inference (Lesson 10).

Illative, that which can be inferred.

Inconsistent terms imply qualities that cannot exist together in the same thing. See Compatible terms.

Indefinite or infinite term is a negative term, which marks out an object only by shutting it out of a class.

Individual, that which cannot be divided without losing its name and distinctive qualities, although it can usually be physically divided, or partitioned; see Partition (Lesson 12).

Intention, first and second, a distinction between terms, defined by Hobbes like this:—"Of the first intention are the names of things, a man, stone, &c.; of the second are the names of names, and speeches, as universal, particular, genus, species, syllogism, and the like." A term of the second intention expresses the way the mind regards or classifies terms of the first intention.

Lemma (λαμβάνω, to take or assume), a proposition, a premise that is granted; in geometry, a preliminary proposition.

Matter is defined by J. S. Mill as "the external cause to which we ascribe our sensations," or as the Permanent Possibility of Sensation.

Membra dividentia, the parts into which a class is divided; the species that make up a genus.

Metaphysics (τὰ μετὰ τὰ Φυσικά), the works of Aristotle that came after his Physics, or were studied after it. Hence First Philosophy, the so-called science of things in their own nature; ontology, or the science of Being.

Method (μέθοδος, μετά and ὁδός, way), a mode, way or instrument for reaching an end.

Metonymy (μετά, and ὄνομα, name), the grammatical name for transferring the meaning of a word to a closely connected thing, as when we say the church and mean the people in it. See Transfer of meaning (Lesson 4).

Necessity (ne, not; and cesso, to cease), that which always is and cannot help but be.

Notion (nosco, to know), the action of grasping or taking note of the various qualities of an object; or, more commonly, the result of that action. See Idea, Concept.

Objective, that which belongs to the object of thought, the non-ego (the not-self); the opposite of Subjective, which see.

Obversion is the name Professor Alexander Bain uses for the process previously called Immediate Inference by Privative Conception (Lesson 10).

Organon (ὄργανον, Latin Organum, Instrument), a name for Aristotle's logical treatises, first in general use in the 15th century. It implies that they can be regarded as an instrument to help the mind. Bacon adopted the name for his Novum Organum.

Paradox (παρά, δόξα, contrary to opinion), an assertion that goes against common opinion, and which may or may not turn out to be true; often wrongly used to mean something self-contradictory and absurd.

Paralogism (παραλογίζομαι, to reason wrongly), a purely logical fallacy, that is, a breach of the rules of deductive logic.

Parity of reasoning, an expression meaning that once one case has been demonstrated, other similar cases can be demonstrated by the same line of reasoning.

Peripatetic Philosophy (περιπατέω, to walk about), the name usually given to the doctrines of Aristotle and his followers, who are said to have carried on their studies and discussions while walking about the halls and walkways of the Lyceum.

Physical definition gives the parts into which a thing can be separated by partition, or physical division.

Polylemma, an argument with the same form as a dilemma, but with more than two alternatives.

Postulate (postulatum, a thing demanded), a proposition that has to be demanded as a basis for argument; in geometry, the postulates lay down the practical conditions required.

Predicaments (prædicamenta, what can be predicated), see Categories.

Principle (principium, beginning), the first source of anything; sometimes used specially to mean the major premise of a syllogism.

Probability, quantity or degree of belief, or, more truly, quantity of information about an uncertain event. It is measured by the ratio of the number of cases favorable to the event to the total number of possible cases.

Problem (πρόβλημα, that which is thrown down), an assertion put forward to be proved or disproved.

Proof, giving a reason or argument in support of a given proposition.

Ratiocination, a name equivalent to Syllogism or Deduction, adopted by J. S. Mill.

Reason (ratio, from reor, to think), a term with a wide and ambiguous meaning; it has sometimes been used specially for the minor premise of a syllogism.

Reductio ad absurdum or ad impossibile, an indirect demonstration based on the impossibility of a contradictory supposition.

Relation (relatum, past participle of refero, to bear back), any connection, in thought or in fact, between two things.

Scholastic Philosophy, a general name for the systems of philosophy taught in the Middle Ages, from the 9th to the 16th century, at their height chiefly in the 13th and 14th centuries. The subject was mainly the logic of Aristotle, mixed with theology, metaphysics, grammar or rhetoric.

Secundi adjacentis of the second adjacent, an expression in incorrect Latin, applied to a grammatical sentence or proposition that has only two parts, the subject and the verb, with no separate copula.

Significates of a term are the things it denotes or signifies.

Sophism (σόφισμα, from σοφία, wisdom), a false argument; the name often implies that the false argument is being used knowingly, to deceive.

Subjective, that which belongs to the thinking subject, the ego, or the mind engaged in thought; the opposite of Objective, which see.

Substance (sub, under; stans from stare, to stand), that which underlies and supports phenomena or attributes. Strictly speaking it is either mind or matter, but it is more commonly used in the material sense.

Subsumption (sub, under; sumo, to take or put), Sir William Hamilton's name for the minor premise of a syllogism, because it brings, or subsumes, a special case under the rule stated in the major premise, or sumption.

Subsumption of a law is Mr. Mill's expression for the third way of explaining a law: by showing it to be a particular case of a more general law.

Sumption (sumo, to take), Hamilton's name for the major premise of a syllogism.

Synthetic syllogism, a syllogism in which the conclusion comes last; see Analytic syllogism.

System (σύστημα, from συνίστημι, to put together), a connected body of knowledge.

Tertii adjacentis of the third adjacent, an expression in incorrect Latin, applied to a grammatical sentence or proposition in which the subject, copula and predicate are all stated separately.

Theory (θεωρία, contemplation), knowledge of principles, as opposed to practice; used ambiguously.

Thesis (θέσις, from τίθημι, to place), an assertion or proposition put forward to be proved or supported by arguments.

Totum divisum, a class or notion that is divided into parts by a difference.

Trilemma, an argument like a dilemma, but with three alternatives.

Truth, the agreement of our knowledge with the things known.